Convergence issues in Relational Concept Analysis based on AOC-posets

# Convergence issues in Relational Concept Analysis based on AOC-posets
Source: [https://arxiv.org/html/2609.00054](https://arxiv.org/html/2609.00054)
Xavier DolquesAgnès BraudAddress:Univ\. de Strasbourg, ENGEES, CNRS, ICube UMR 7357, F\-67000 Strasbourg, FranceAlain GutierrezAddress:LIRMM, Univ\. Montpellier, CNRS, Montpellier, FranceMarianne HuchardEmail:[marianne\.huchard@lirmm\.fr](mailto:[email protected])Corresponding author:Corresponding authorAddress:LIRMM, Univ\. Montpellier, CNRS, Montpellier, FranceFlorence Le BerAddress:Univ\. de Strasbourg, ENGEES, CNRS, ICube UMR 7357, F\-67000 Strasbourg, France

###### Abstract

Formal Concept Analysis \(FCA\) is an approach for conceptual classification building and rule discovery from a binary table describing a set of objects by a set of attributes\. Extensions have been proposed to deal with non\-binary and more complex data, such as Relational Concept Analysis \(RCA\) for multi\-relational data\. RCA aims to highlight groups of objects characterized by their relationships with other groups of objects\. The richer and more complex nature of the underlying data allows RCA to produce richer results than FCA, at the expense of higher computational and interpretive complexity\. The most commonly used conceptual classification structure in FCA is the concept lattice\. However, in many applications, concept\-lattice substructures—such as AOC\-posets—are preferred over the full lattice, either to mitigate combinatorial blow\-up or to focus on the most informative parts of the structure\. Indeed, in an AOC\-poset, only concepts introducing an object or an attribute are represented, which makes AOC\-posets smaller and easier to compute and use than concept lattices\. Although RCA was originally defined on concept lattices, it can also be instantiated on AOC\-posets\. RCA is iterative and its convergence is guaranteed in the lattice\-based setting, but this guarantee is lost when using AOC\-posets\. In this paper, we investigate this loss of convergence in detail\. We show why convergence is no longer guaranteed in the general case, identify conditions under which it can still be ensured, and discuss how a dataset can be transformed to recover convergence\. We also propose a convergent variant of the process, which preserves the AOC\-poset structure: relational attributes, once created, are never removed, which guarantees convergence at the price of attributes that may refer to concepts absent from the final structures\.

###### Keywords:

Formal Concept Analysis , Concept lattice , Relational Concept Analysis , AOC\-posets , convergence

## 1Introduction

Formal Concept Analysis \(FCA\)\([Ganter and Wille, 1999](https://arxiv.org/html/2609.00054#bib.bib36)\)provides a natural framework when the goal of data analysis is not primarily prediction, but the discovery and organization of interpretable and explainable knowledge patterns, in line with a white\-box perspective on knowledge representation\([Atzmueller et al\., 2024](https://arxiv.org/html/2609.00054#bib.bib3)\)\. Based on lattice theory, FCA aims to discover conceptual structures from sets of objects and their attributes\. It identifies formal concepts, each defined by a set of objects and the attributes they share, and organizes them into a concept lattice that makes abstraction, specialization, and dependency relations explicit\. This contrasts with many mainstream machine\-learning approaches, including supervised learning, clustering, dimensionality\-reduction techniques, and neural models, which often focus on predictive accuracy, numerical similarity, or compact numerical representations\([Hastie et al\., 2009](https://arxiv.org/html/2609.00054#bib.bib1);[LeCun et al\., 2015](https://arxiv.org/html/2609.00054#bib.bib2)\)\. While explainable AI often aims at providing post\-hoc explanations for predictive black\-box models\([Guidotti et al\., 2018](https://arxiv.org/html/2609.00054#bib.bib4)\), FCA offers an intrinsically interpretable symbolic representation and synthetic representation of the data\. It supports knowledge discovery by producing explicit conceptual structures, abstract relations, and implications that can be inspected, interpreted, and discussed by domain experts\. FCA has been successfully used for analyzing tabular data, either binary or multivalued\([Poelmans et al\., 2013a](https://arxiv.org/html/2609.00054#bib.bib58);[Poelmans et al\., 2013b](https://arxiv.org/html/2609.00054#bib.bib59)\)\. Various extensions have later been proposed to process more complex data, such as multidimensional data\([Voutsadakis, 2002](https://arxiv.org/html/2609.00054#bib.bib66)\), fuzzy data\([Belohlávek and Vychodil, 2005](https://arxiv.org/html/2609.00054#bib.bib17)\), and relational data\([Rouane\-Hacène et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib60);[Ferré and Cellier, 2020](https://arxiv.org/html/2609.00054#bib.bib33);[Kötters and Eklund, 2020](https://arxiv.org/html/2609.00054#bib.bib47)\)\.

Relational Concept Analysis \(RCA\)\([Rouane\-Hacène et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib60)\)is one of the extensions to handle relational data\. In this framework, objects are described by attributes, and by relationships between them\. RCA is based on the iterative use of FCA, computing several concept lattices, which are connected by links that abstract the relations between objects\. The result complements graph\-pattern\-based analyses\([Ferré and Cellier, 2020](https://arxiv.org/html/2609.00054#bib.bib33);[Kötters and Eklund, 2020](https://arxiv.org/html/2609.00054#bib.bib47)\)by highlighting how objects are classified separately within each category according to these relations\. Another distinctive feature of RCA is its use of several scaling operators borrowed from Description Logics\([Baader et al\., 2003](https://arxiv.org/html/2609.00054#bib.bib15)\)to build links between concepts\. This contrasts with approaches restricted to the existential operator, such as[Ferré and Cellier \(2020\)](https://arxiv.org/html/2609.00054#bib.bib33);[Kötters and Eklund \(2020\)](https://arxiv.org/html/2609.00054#bib.bib47), and enables the extraction of richer patterns\. Nevertheless, since RCA groups objects using relationships to objects at any distance, it often comes with a combinatorial explosion, and patterns of interest are difficult to extract from the huge set of built concepts\. Note that this is true for other FCA\-based methods for relational data\. Various strategies can be used in RCA to cope with this complexity, including separating the initial formal object sets into smaller ones after a first analysis, introducing queries\([Azmeh et al\., 2011b](https://arxiv.org/html/2609.00054#bib.bib14)\), using frequency thresholds or specific measures to select the concepts of interest\([Stumme et al\., 2002](https://arxiv.org/html/2609.00054#bib.bib65);[Buzmakov et al\., 2014](https://arxiv.org/html/2609.00054#bib.bib8);[Kuznetsov and Makhalova, 2018](https://arxiv.org/html/2609.00054#bib.bib5)\), or using an alternative conceptual structure—namely, an AOC\-poset—as a substitute for the concept lattices\([Dolques et al\., 2013b](https://arxiv.org/html/2609.00054#bib.bib28)\)\. AOC\-posets are sub\-orders of lattices restricted to concepts introducing objects or attributes, called respectively object\-concepts \(OC\) and attribute\-concepts \(AC\)\. They are smaller and computationally more efficient than concept lattices while preserving the original information\([Godin and Mili, 1993](https://arxiv.org/html/2609.00054#bib.bib37)\)\.

Using AOC\-posets for RCA was motivated by the analysis of particular relational datasets, e\.g\. in environmental data\([Dolques et al\., 2016](https://arxiv.org/html/2609.00054#bib.bib29);[Braud et al\., 2022](https://arxiv.org/html/2609.00054#bib.bib21)\), where we showed that this approach provided a reasonable number of concepts and relevant rules\. A second advantage is that the object\- and attribute\-concepts contained in AOC\-posets may be the only outputs needed in some applications\. This is the case, for instance, in software engineering tasks such as class model refactoring\([Miralles et al\., 2015](https://arxiv.org/html/2609.00054#bib.bib50)\)\. In this setting, the goal is to derive more general classes from an initial class model in which generalization relations are incomplete or missing\. Attribute\-concepts can guide the construction of these generalized classes, while object\-concepts help preserve the classes of the initial model\. From these experiments, one may suggest that RCA\-AOC has strong potential, both for focusing on the most relevant patterns in the context of a given application and for limiting the amount of extracted knowledge\. Nevertheless, in order to generalize the use of an AOC\-poset\-based RCA process \(RCA\-AOC\) to other applications that rely on different—and potentially more complex—data schemas, it is important to verify whether RCA’s original properties are preserved\. Unfortunately, we found that using AOC\-posets within RCA can introduce divergence issues\. A critical issue then arises: understanding when and why divergences occur, and devising ways either to eliminate them or to reliably exploit the results in their presence\.

In this paper, we illustrate the potential risk of process divergence through three examples involving the two most widely used scaling operators, namely existential scaling and strict universal scaling\. One of these examples is concrete and is grounded in a UML class\-model refactoring task\. Furthermore, we identify properties that prevent divergence in most cases\. We also propose a variant of the process that guarantees termination, builds genuine AOC\-posets, and yields concepts required in certain applications, such as the UML class\-model refactoring task presented in this paper\.

The rest of the paper is organized as follows\. Section[2](https://arxiv.org/html/2609.00054#S2)introduces the basic definitions of RCA\. Section[3](https://arxiv.org/html/2609.00054#S3)details RCA\-AOC, the RCA process based on AOC\-posets\. Section[4](https://arxiv.org/html/2609.00054#S4)presents three examples of process divergence, including one drawn from a real\-world software engineering application\. In Sect\.[5](https://arxiv.org/html/2609.00054#S5), we discuss different approaches to ensuring convergence in the AOC\-poset\-based RCA process, including conditions on the data and on the process itself, as well as a convergent variant of the process\. Section[6](https://arxiv.org/html/2609.00054#S6)presents the state of the art about AOC\-posets and RCA, including theoretical developments, variants, and applications\. Finally, Sect\.[7](https://arxiv.org/html/2609.00054#S7)concludes the paper and outlines directions for future work\.

## 2Introduction to Relational Concept Analysis

In this section, we introduce the RCA process and show the graphical form of its results\. Definitions are illustrated with an example taken from theCNRS Miti’80 2021 Paradiseproject\([Fokou et al\., 2024](https://arxiv.org/html/2609.00054#bib.bib35)\)\.

### 2\.1FCA basics

RCA is a relational variant of Formal Concept Analysis \(FCA\)\([Ganter and Wille, 1999](https://arxiv.org/html/2609.00054#bib.bib36)\)\. FCA takes as input a dataset, called a formal context, composed of objects described by attributes\. Table[1](https://arxiv.org/html/2609.00054#S2.T1)shows a formal context, denoted𝒦Plants\{\\mathcal\{K\}\}\_\{\\textit\{Plants\}\}, which describes plants used in ancient remedies and their useful parts or characteristics\. For example, leaves and roots ofArchangelica officinalis\(a​n​g​e​l​i​c​aangelica\), an herbaceous plant, can be used in remedies\. The aim of FCA is to extract an ordered set of concepts from the input dataset\.

Table 1:Formal Context of plants and their used parts \(𝒦P​l​a​n​t​s\{\\mathcal\{K\}\}\_\{Plants\}\)###### Definition 1\(Formal Context\)\.

A formal context is a triple𝒦=\(G,M,I\)\{\\mathcal\{K\}\}=\(G,M,I\)whereGGandM\{M\}are finite sets of objects and attributes, respectively, andI\{I\}is the incidence relation, i\.e\.,I⊆G×M\{I\}\\penalty\\ \\subseteq\\penalty\\ \{G\}\\times\{M\}\.

###### Definition 2\(Derivation operators\)\.

Let𝒦=\(G,M,I\)\\mathcal\{K\}=\(G,M,I\)be a formal context\. The two derivation operators, both denoted by\(⋅\)′\(\\cdot\)^\{\\prime\}, are defined, forX⊆GX\\subseteq GandY⊆MY\\subseteq M, by:

X′=\{m∈M∣∀g∈X,\(g,m\)∈I\}Y′=\{g∈G∣∀m∈Y,\(g,m\)∈I\}X^\{\\prime\}=\\\{m\\in M\\mid\\forall g\\in X,\\ \(g,m\)\\in I\\\}\\qquad Y^\{\\prime\}=\\\{g\\in G\\mid\\forall m\\in Y,\\ \(g,m\)\\in I\\\}X′X^\{\\prime\}is the set of attributes shared by all objects ofXX, andY′Y^\{\\prime\}the set of objects owning all attributes ofYY\. Composing the two operators yields the closure operators\(⋅\)′′\(\\cdot\)^\{\\prime\\prime\}onGGand onMM\. For a single objecto∈Go\\in G, we write\{o\}′\\\{o\\\}^\{\\prime\}and\{o\}′′\\\{o\\\}^\{\\prime\\prime\}\.

###### Definition 3\(Formal Concept\)\.

A formal concept of𝒦=\(G,M,I\)\\mathcal\{K\}=\(G,M,I\)is a pairC=\(X,Y\)C=\(X,Y\)withX⊆GX\\subseteq GandY⊆MY\\subseteq M, such thatX′=YX^\{\\prime\}=YandY′=XY^\{\\prime\}=X\.X=E​x​t​e​n​t​\(C\)X=Extent\(C\)is the extent of the concept, i\.e\., the set of objects falling under the concept, andY=I​n​t​e​n​t​\(C\)Y=Intent\(C\)is its intent, i\.e\., the set of attributes shared by these objects\.

###### Definition 4\(Concept Specialization Order and Lattice\)\.

Let𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}be the set of all concepts built from formal context𝒦\\mathcal\{K\}\. LetC1=\(X1,Y1\)C\_\{1\}=\(X\_\{1\},Y\_\{1\}\)andC2=\(X2,Y2\)C\_\{2\}=\(X\_\{2\},Y\_\{2\}\)be two elements of𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\. The concept specialization order≤s\\leq\_\{s\}is defined byC1≤sC2C\_\{1\}\\leq\_\{s\}C\_\{2\}if and only ifX1⊆X2X\_\{1\}\\subseteq X\_\{2\}\(and equivalentlyY2⊆Y1Y\_\{2\}\\subseteq Y\_\{1\}\)\.C1C\_\{1\}is called a subconcept ofC2C\_\{2\}, whileC2C\_\{2\}is called a superconcept ofC1C\_\{1\}\. The concept set𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}provided with the specialization order, \(𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\},≤s\\leq\_\{s\}\), has a lattice structure, and is called the concept lattice associated with𝒦\{\\mathcal\{K\}\}\.

The concept lattice built on𝒦Plants\{\\mathcal\{K\}\}\_\{\\textit\{Plants\}\}is shown in Fig\.[1](https://arxiv.org/html/2609.00054#S2.F1)\. To reduce redundant information in the presentation of concept lattices, figures only show introduced objects \(resp\. introduced attributes\) in concepts\. E\.g\.C\_plants\_8\(denotedcp\_8for short\) introduces attributef​l​o​w​e​r​sflowersand objectc​a​m​o​m​i​l​ecamomile111C\_plants\_8is the introducer concept of attributef​l​o​w​e​r​sflowersand objectc​a​m​o​m​i​l​ecamomile\.\. But the complete concept definition isI​n​t​e​n​t​\(𝚌𝚙​\_​𝟾\)Intent\(\\mathtt\{cp\\\_8\}\)=\{herbaceous\\\{herbaceous,flowers\}flowers\\\}withh​e​r​b​a​c​e​o​u​sherbaceoustop\-down inherited, andE​x​t​e​n​t​\(CLOSEExtent\(OPEN𝚌𝚙​\_​𝟾\)\\mathtt\{cp\\\_8\}\)=\{camomile\\\{camomile,b​o​r​a​g​eborage,garlic\}garlic\\\}withb​o​r​a​g​eborageandg​a​r​l​i​cgarlicbottom\-up inherited\. The presentation exploits the fact that when an attribute belongs to the intent of a conceptCC, it is \(top\-down\) inherited byCCsubconcepts\. Thus it is sufficient to show an attribute in the highest concept having it in its intent\. This highest concept is thei​n​t​r​o​d​u​c​i​n​gintroducingconcept of the attribute\. The same simplification can be symmetrically applied to the objects, which are bottom\-up inherited\.

Figure 1:Concept Lattice on𝒦Plants\{\\mathcal\{K\}\}\_\{\\textit\{Plants\}\}\. Each concept is represented in 3 parts: its identifier \(top\), its intent \(middle\), and its extent \(bottom\); only introduced attributes and objects are shown in intents and extents respectively\. In the text,C\_plants\_iis shortened tocp\_i\.
### 2\.2RCA process

Relational Concept Analysis \(RCA\) aims at extending FCA to take into account datasets where objects of several categories are described by attributes and by relations to objects\([Huchard et al\., 2007](https://arxiv.org/html/2609.00054#bib.bib46);[Rouane\-Hacène et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib60)\)\. The main idea of RCA is as follows: when objects of a category are connected via a relation to objects of another category \(or the same category\), concepts formed on top of objects of the latter category can be used to form concepts on top of objects of the former category\. The whole dataset is called a Relational Context Family\.

###### Definition 5\(Relational Context Family \(RCF\)\)\.

A Relational Context Family \(denoted RCF\) is a\(𝐊,𝐑\)\(\{\\mathbf\{K\}\},\{\\mathbf\{R\}\}\)pair where:

- 1\.𝐊=\{𝒦i\}i=1,…,n\{\\mathbf\{K\}\}=\\\{\{\\cal K\}\_\{i\}\\\}\_\{i=1,\\ldots,n\}is a set of contexts,𝒦i=\(Gi,Mi,Ii\)\{\\cal K\}\_\{i\}=\(G\_\{i\},M\_\{i\},I\_\{i\}\)\.
- 2\.𝐑=\{rj\}j=1,…,m\{\\mathbf\{R\}\}=\\\{r\_\{j\}\\\}\_\{j=1,\\ldots,m\}is a set of relations,rj⊆Gsrj×Gtrjr\_\{j\}\\subseteq G\_\{s\_\{r\_\{j\}\}\}\\times G\_\{t\_\{r\_\{j\}\}\}wheresrj,trj∈\{1,…,n\}s\_\{r\_\{j\}\},t\_\{r\_\{j\}\}\\in\\\{1,\\ldots,n\\\}, andGsrjG\_\{s\_\{r\_\{j\}\}\}andGtrjG\_\{t\_\{r\_\{j\}\}\}are resp\. the source \(domain\) and target \(range\) ofrjr\_\{j\}\.

In the following, we assume the object sets \(GiG\_\{i\},i=1,…,ni=1,\.\.\.,n\) to be pairwise disjoint\. The relationship between objects of a category and concepts formed on top of objects of another category is implemented thanks toscaling operators, which are inspired by operators of Description Logics\. This results in the creation of special attributes calledrelational attributes, of the formq​r​\(C\)qr\(C\), whereqqis a scaling operator,rra relation between two object setsGsrG\_\{s\_\{r\}\}andGtrG\_\{t\_\{r\}\}, andCCa concept from the lattice built on the context𝒦tr=\(Gtr,Mtr,Itr\)\{\\cal K\}\_\{t\_\{r\}\}=\(G\_\{t\_\{r\}\},M\_\{t\_\{r\}\},I\_\{t\_\{r\}\}\)\. The most used scaling operators are:

- 1\.Theexistentialscaling operator \(∃\\exists\): an objecto∈Gsro\\in G\_\{s\_\{r\}\}is in relation by∃r\\exists rwith a conceptCCfrom𝒦tr\{\\cal K\}\_\{t\_\{r\}\}ifr⁡\(o\)r\(o\)has a non\-empty intersection withE​x​t​e​n​t​\(C\)Extent\(C\), wherer⁡\(o\)r\(o\)denotes the image set ofoobyrr, i\.e\.oois connected viarrto at least one element fromE​x​t​e​n​t​\(C\)Extent\(C\)\. A relational attribute based on this scaling operator is denoted as∃r⁡\(C\)\\exists r\(C\)\.
- 2\.Theuniversalscaling operator \(∀\\forall\): an objecto∈Gsro\\in G\_\{s\_\{r\}\}is in relation by∀r\\forall rwith a conceptCCfrom𝒦tr\{\\cal K\}\_\{t\_\{r\}\}ifr⁡\(o\)r\(o\)is included in the extent ofCC, i\.e\.oois connected viarronly to elements fromE​x​t​e​n​t​\(C\)Extent\(C\)\. A relational attribute based on this scaling operator is denoted as∀r⁡\(C\)\\forall r\(C\)\.
- 3\.Thestrict universalscaling operator \(∃∀\\exists\\forall\): an objecto∈Gsro\\in G\_\{s\_\{r\}\}is in relation by∃∀⁡r\\exists\\forall rwith a conceptCCfrom𝒦tr\{\\cal K\}\_\{t\_\{r\}\}ifr⁡\(o\)r\(o\)isnon\-emptyand included in the extent ofCC, i\.e\.oois connected viarronly to elements fromE​x​t​e​n​t​\(C\)Extent\(C\)\(andat least one\)\. A relational attribute based on this scaling operator is denoted as∃∀⁡r⁡\(C\)\\exists\\forall r\(C\)\.

When processing an RCF\(𝐊,𝐑\)\(\{\\mathbf\{K\}\},\{\\mathbf\{R\}\}\)with RCA, different scaling operators can be applied to the relations of𝐑\{\\mathbf\{R\}\}\. We introduce then the functionρ:𝐑→\{∃,∀,∃∀\}\\rho:\{\\mathbf\{R\}\}\\rightarrow\\\{\\exists,\\forall,\\exists\\forall\\\}that associates a scaling operator to a relation from𝐑\{\\mathbf\{R\}\}\.

We here explain these principles relying on an example taken from a database on ancient Arabic remedies\([Fokou et al\., 2024](https://arxiv.org/html/2609.00054#bib.bib35)\)\. The relational context family, calledRPin the following, is composed of a Plants context,𝒦Plants\{\\mathcal\{K\}\}\_\{\\textit\{Plants\}\}\(see Table[1](https://arxiv.org/html/2609.00054#S2.T1)\), a Remedies context, denoted𝒦Remedies\{\\mathcal\{K\}\}\_\{\\textit\{Remedies\}\}\(Table at the left\-hand side of Fig\.[2](https://arxiv.org/html/2609.00054#S2.F2)\), and acontainsrelation, denotedrc​o​n​t​a​i​n​s\{r\}\_\{contains\}\(Table[2](https://arxiv.org/html/2609.00054#S2.T2)\)\. Remedies are described by their ingredients and their application form; two forms are here considered: pills and potions\. Let us for example consider the concept lattice of Fig\.[1](https://arxiv.org/html/2609.00054#S2.F1)and thecp\_8concept which groups plants whose flowers are used in remedies, namelycamomile,borageandgarlic\. Now, according to the existential scaling operator,remedy1,remedy2andremedy3can be linked tocp\_8because they contain at least one plant with useful flowers, i\.e\. at least one plant inE​x​t​e​n​t​\(CLOSEExtent\(cp\_8\)\)\(Table[2](https://arxiv.org/html/2609.00054#S2.T2)\)\. To represent this property, a new attribute is added to objectsremedies\. This is done for each concept from𝒦Plants\{\\mathcal\{K\}\}\_\{\\textit\{Plants\}\}leading to an extension of the original context, as defined below\.

Figure 2:Formal Context𝒦Remedies\{\\mathcal\{K\}\}\_\{\\textit\{Remedies\}\}\(left\) and concept lattice \(right\) for remedies\. In the text,C\_remedies\_iis shortened tocr\_i\.Table 2:Relationrc​o​n​t​a​i​n​s\\mathit\{r\}\_\{contains\}###### Definition 6\(Existential scaling \(∃\\exists\)\)\.

Let𝒦=\(G,M,I\)\{\\cal K\}=\(G,M,I\)be a context in a relational context family, andrra relation, whereG=GsrG=G\_\{s\_\{r\}\}is the source ofrr, andGtrG\_\{t\_\{r\}\}, the target ofrr, is the object set of a formal context𝒦tr=\(Gtr,Mtr,Itr\)\{\\cal K\}\_\{t\_\{r\}\}=\(G\_\{t\_\{r\}\},M\_\{t\_\{r\}\},I\_\{t\_\{r\}\}\)\. Let also𝒞tr\{\\cal C\}\_\{t\_\{r\}\}be the set of concepts of the lattice built on𝒦tr\{\\cal K\}\_\{t\_\{r\}\}\. The application of𝕊∃\\mathbb\{S\}\_\{\\exists\}, the existential scaling operator, to context𝒦\{\\cal K\}, denoted by𝕊∃\\mathbb\{S\}\_\{\\exists\}\(𝒦,r,𝒞tr\{\\cal K\},r,\{\\cal C\}\_\{t\_\{r\}\}\), gives the extension𝒦\+=\(G\+,M\+,I\+\)\{\\cal K\}^\{\+\}=\(G^\{\+\},M^\{\+\},I^\{\+\}\)of𝒦\{\\cal K\}, with:

- 1\.G\+=GG^\{\+\}=G
- 2\.M\+=\{∃r⁡\(C\)\|C∈𝒞tr\}M^\{\+\}=\\\{\\exists r\(C\)\\penalty\\ \|\\penalty\\ C\\in\{\\cal C\}\_\{t\_\{r\}\}\\\}\.
- 3\.I\+=\{\(o,∃r\(C\)\)\|o∈G,C∈𝒞tr,r\(o\)∩Extent\(C\)≠∅\}I^\{\+\}=\\\{\(o,\\exists r\(C\)\)\\penalty\\ \|\\penalty\\ o\\in G,C\\in\{\\cal C\}\_\{t\_\{r\}\},r\(o\)\\cap Extent\(C\)\\neq\\emptyset\\\}

Table[3](https://arxiv.org/html/2609.00054#S2.T3)shows𝕊∃​\(KRemedies,rc​o​n​t​a​i​n​s,𝒞Plants\)\\mathbb\{S\}\_\{\\exists\}\(K\_\{\\textit\{Remedies\}\},r\_\{contains\},\{\\cal C\}\_\{\\textit\{Plants\}\}\)where𝒞Plants\{\\cal C\}\_\{\\textit\{Plants\}\}is the set of concepts of the lattice of Fig\.[1](https://arxiv.org/html/2609.00054#S2.F1)\. In this case,I\+I^\{\+\}contains e\.g\.\(remedy1CLOSE,\(\\texttt\{remedy1\},OPEN∃rc​o​n​t​a​i​n​s​\(cp\_8\)\)\\exists\\penalty\\ \{r\_\{contains\}\}\(\\texttt\{cp\\\_8\}\)\),\(remedy2CLOSE,\(\\texttt\{remedy2\},OPEN∃rc​o​n​t​a​i​n​s​\(cp\_8\)\)\\exists\\penalty\\ \{r\_\{contains\}\}\(\\texttt\{cp\\\_8\}\)\), and\(remedy3CLOSE,\(\\texttt\{remedy3\},∃rc​o​n​t​a​i​n​s\\exists\\penalty\\ \{r\_\{contains\}\}OPEN\(cp\_8\)\)\(\\texttt\{cp\\\_8\}\)\), becauseremedy1containsc​a​m​o​m​i​l​ecamomile,remedy2andremedy3containg​a​r​l​i​cgarlic, both plants incp\_8extent\. This extension is added to the original context and then used to update the Remedies lattice\. The new concept lattice is shown in Fig\.[3](https://arxiv.org/html/2609.00054#S2.F3)\.

Table 3:Existential scaling of the formal context for Remedies based on the relationrc​o​n​t​a​i​n​sr\_\{contains\}\(shortened torrin the column headers\),𝕊∃​\(𝒦Remedies,r𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠,𝒞Plants\)\\mathbb\{S\}\_\{\\exists\}\(\\mathcal\{K\}\_\{\\textit\{Remedies\}\},\\mathit\{r\_\{contains\}\},\\mathcal\{C\}\_\{\\textit\{Plants\}\}\)\.###### Definition 7\(Strict universal scaling \(∃∀\\exists\\forall\)\)\.

Let𝒦=\(G,M,I\)\{\\cal K\}=\(G,M,I\)be a context in a relational context family, andrra relation, whereG=GsrG=G\_\{s\_\{r\}\}is the source ofrr, andGtrG\_\{t\_\{r\}\}, the target ofrr, is the object set of a formal context𝒦tr=\(Gtr,Mtr,Itr\)\{\\cal K\}\_\{t\_\{r\}\}=\(G\_\{t\_\{r\}\},M\_\{t\_\{r\}\},I\_\{t\_\{r\}\}\)\. Let also𝒞tr\{\\cal C\}\_\{t\_\{r\}\}be the set of concepts of the lattice built on𝒦tr\{\\cal K\}\_\{t\_\{r\}\}\. The application of𝕊∃∀\\mathbb\{S\}\_\{\\exists\\forall\}, the strict universal scaling operator, to context𝒦\{\\cal K\}, denoted by𝕊∃∀\\mathbb\{S\}\_\{\\exists\\forall\}\(𝒦,r,𝒞tr\{\\cal K\},r,\{\\cal C\}\_\{t\_\{r\}\}\), gives the extension𝒦\+=\(G\+,M\+,I\+\)\{\\cal K\}^\{\+\}=\(G^\{\+\},M^\{\+\},I^\{\+\}\), with:

- 1\.G\+=GG^\{\+\}=G
- 2\.M\+=\{∃∀⁡r⁡\(C\)\|C∈𝒞tr\}M^\{\+\}=\\\{\\exists\\forall r\(C\)\\penalty\\ \|\\penalty\\ C\\in\{\\cal C\}\_\{t\_\{r\}\}\\\}
- 3\.I\+=\{\(o,∃∀r\(C\)\)\|o∈G,C∈𝒞tr,r\(o\)≠∅𝑎𝑛𝑑r\(o\)⊆Extent\(C\)\}I^\{\+\}=\\\{\(o,\\exists\\forall r\(C\)\)\\penalty\\ \|\\penalty\\ o\\in G,C\\in\{\\cal C\}\_\{t\_\{r\}\},r\(o\)\\neq\\emptyset\\,\\mathit\{and\}\\,r\(o\)\\subseteq Extent\(C\)\\\}

###### Definition 8\(Relational extension of a context\)\.

Let𝒦=\(G,M,I\)\{\\cal K\}=\(G,M,I\)be a context in a relational context family\(𝐊,𝐑\)\(\{\\mathbf\{K\}\},\{\\mathbf\{R\}\}\), andRGR\_\{G\}the subset of relationsr∈𝐑r\\in\\mathbf\{R\}such thatGsr=GG\_\{s\_\{r\}\}=G\. The relational extension of𝒦\{\\cal K\}, denoted𝔼ρ,𝐂​\(𝒦\)\\mathbb\{E\}\_\{\\rho,\{\\mathbf\{C\}\}\}\(\{\\cal K\}\), is the result of the apposition \(denoted by the symbol ‘\|’\) of𝒦\{\\cal K\}with all extensions𝕊ρ⁡\(r\)\\mathbb\{S\}\_\{\\rho\(r\)\}\(𝒦,r,𝒞tr\{\\cal K\},r,\{\\cal C\}\_\{t\_\{r\}\}\) forr∈RGr\\in R\_\{G\}, and𝐂=\\mathbf\{C\}=⋃r∈RG𝒞tr\\bigcup\_\{r\\in R\_\{G\}\}\{\\mathcal\{C\}\_\{t\_\{r\}\}\}\.

𝔼ρ,𝐂​\(𝒦\)=𝒦\|𝕊ρ⁡\(r1\)​\(𝒦,r1,𝒞tr1\)​\|…\|​𝕊ρ⁡\(rk\)​\(𝒦,rk,𝒞trk\)\\mathbb\{E\}\_\{\\rho,\{\\mathbf\{C\}\}\}\(\{\\cal K\}\)=\{\\cal K\}\\penalty\\ \|\\penalty\\ \\mathbb\{S\}\_\{\\rho\(r\_\{1\}\)\}\(\{\\cal K\},r\_\{1\},\{\\cal C\}\_\{t\_\{r\_\{1\}\}\}\)\\penalty\\ \|\\penalty\\ \\ldots\|\\penalty\\ \\mathbb\{S\}\_\{\\rho\(r\_\{k\}\)\}\(\{\\cal K\},r\_\{k\},\{\\cal C\}\_\{t\_\{r\_\{k\}\}\}\)

By extension,Eρ∗​\(𝐊\)E^\{\*\}\_\{\\rho\}\(\\mathbf\{K\}\)denotes the relational extension of𝐊\\mathbf\{K\}, which is composed of all the relational extensions of all𝒦i\\mathcal\{K\}\_\{i\}in𝐊\\mathbf\{K\}:

Eρ∗​\(𝐊\)=\{Eρ,𝐂1​\(𝒦1\),…,Eρ,𝐂n​\(𝒦n\)\}E^\{\*\}\_\{\\rho\}\(\\mathbf\{K\}\)=\\\{E\_\{\\rho,\{\\mathbf\{C\}\_\{1\}\}\}\(\\mathcal\{K\}\_\{1\}\),\\ldots,E\_\{\\rho,\{\\mathbf\{C\}\_\{n\}\}\}\(\\mathcal\{K\}\_\{n\}\)\\\}where, for eachii:

𝐂i=⋃r∈RGi𝒞tr\{\\mathbf\{C\}\_\{i\}\}=\\bigcup\_\{r\\in R\_\{G\_\{i\}\}\}\\mathcal\{C\}\_\{t\_\{r\}\}
A relational extension of𝐊=\{𝒦Plants,𝒦Remedies\}\{\\mathbf\{K\}\}=\\\{\{\\mathcal\{K\}\}\_\{\\textit\{Plants\}\},\{\\mathcal\{K\}\}\_\{\\textit\{Remedies\}\}\\\}is composed of Table[1](https://arxiv.org/html/2609.00054#S2.T1)\(no outgoing relation\), and the left table of Fig\.[2](https://arxiv.org/html/2609.00054#S2.F2)apposed to Table[3](https://arxiv.org/html/2609.00054#S2.T3)\. The resulting context is𝐊1=𝔼ρ∗​\(𝐊\)=\{𝒦Plants,𝔼ρ,𝐂​\(𝒦Remedies\)\}\{\\mathbf\{K\}\}^\{1\}=\\mathbb\{E\}^\{\*\}\_\{\\rho\}\(\{\\mathbf\{K\}\}\)=\\\{\{\\mathcal\{K\}\}\_\{\\textit\{Plants\}\},\\mathbb\{E\}\_\{\\rho,\{\\mathbf\{C\}\}\}\(\{\\cal K\}\_\{\\textit\{Remedies\}\}\)\\\}\. Figure[3](https://arxiv.org/html/2609.00054#S2.F3)shows the concept lattice built from the extended context𝔼ρ,𝐂​\(𝒦Remedies\)=𝒦Remedies\|𝕊∃​\(KRemedies,r𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠,𝒞Plants\)\\mathbb\{E\}\_\{\\rho,\{\\mathbf\{C\}\}\}\(\{\\cal K\}\_\{\\textit\{Remedies\}\}\)=\{\\mathcal\{K\}\}\_\{\\textit\{Remedies\}\}\|\\mathbb\{S\}\_\{\\exists\}\(K\_\{\\textit\{Remedies\}\},r\_\{\\mathit\{contains\}\},\{\\cal C\}\_\{\\textit\{Plants\}\}\)\.

Figure 3:Concept Lattice on𝕊∃​\(𝒦Remedies,rc​o​n​t​a​i​n​s,𝒞Plants\)\\mathbb\{S\}\_\{\\exists\}\(\{\\mathcal\{K\}\}\_\{\\textit\{Remedies\}\},r\_\{contains\},\{\\cal C\}\_\{\\textit\{Plants\}\}\)apposed to𝒦Remedies\{\\mathcal\{K\}\}\_\{\\textit\{Remedies\}\}\.A whole construction process consists in building a finite sequence of contexts and concept lattices associated with\(𝐊,𝐑\)\(\{\\mathbf\{K\}\},\{\\mathbf\{R\}\}\)andρ\\rho\. The first set of contexts \(step00\) is𝐊0=𝐊\{\\mathbf\{K\}\}^\{0\}=\{\\mathbf\{K\}\}\. The contexts of stepppare extended with relational attributes built on the concepts of𝐂p−1\{\\mathbf\{C\}\}^\{p\-1\}, i\.e\. the set of concepts of all lattices built at the previous step\. These extended contexts are used to build new lattices providing a new set of concepts𝐂p\{\\mathbf\{C\}\}^\{p\}that will be used in the next step\. The process stops when a fixpoint is obtained, i\.e\. lattices associated with the same formal context at two successive steps have the same sets of concept extents\. The result is a family of interrelated lattices, called aConcept Lattice Family\(CLF\)\.

### 2\.3RCA output as graphs

RCA is usually described as a process, starting from an RCF and producing a family of interrelated concept lattices\([Rouane\-Hacène et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib60)\)\. Nevertheless, several works have highlighted the links between these outputs and graphs, in various domains, e\.g\.[Dolques et al\. \(2010\)](https://arxiv.org/html/2609.00054#bib.bib26)produced model transformation patterns in the software domain\.[Nica et al\. \(2020a\)](https://arxiv.org/html/2609.00054#bib.bib52);[Nica et al\. \(2020b\)](https://arxiv.org/html/2609.00054#bib.bib53)proposed a general method, called RCA\-Seq, to extract a hierarchy of directed acyclic graphs from the lattice family, starting from one chosen lattice\. In the resulting hierarchy, each graph represents a navigation path through the lattice family\. Redundant links have been removed\. In the same way,[Ferré and Cellier \(2018\)](https://arxiv.org/html/2609.00054#bib.bib32)introduced a generic representation of RCA outputs as a hierarchy of concept graphs where each concept of the lattice family belongs to one concept graph and each concept graph exhibits the relationships between several concepts\.

Following this idea, a recent work has shown that RCA outputs can be represented as a set of relational patterns with no loss of information\([Fokou et al\., 2025](https://arxiv.org/html/2609.00054#bib.bib34)\)\. These relational patterns are directly comparable to graph patterns produced by the Graph\-FCA approach\([Ferré and Cellier, 2020](https://arxiv.org/html/2609.00054#bib.bib33)\)when inverse relations are included in the relational context family\. The set of relational patterns is derived from a dependency graphG=\(V,E\)G=\(V,E\), whereVVis the set of all concepts described with unary attributes,EEis the set of dependencies \(relational attributes and subsumption\) between concepts\. This graph is built on top of the concept lattice family, by extracting maximal subsets of related concepts and removing redundant links\. This approach cannot be directly applied on RCF with one\-way relations\. But the approach proposed by[Nica et al\. \(2020b\)](https://arxiv.org/html/2609.00054#bib.bib53)for temporal data can be adapted for any data model of a relational context family\.

![Refer to caption](https://arxiv.org/html/2609.00054v1/Remedes__pharmaco_RCA-patterns-revu.png)

![Refer to caption](https://arxiv.org/html/2609.00054v1/Remedes__pharmaco_RCA-patternsrevu-focus0.png)

Figure 4:General overview of the patterns extracted from the concept lattice family built onRPwith the existential quantifier and a focus oncr\_0and linked nodes\.As formalized by[Nica et al\. \(2020b\)](https://arxiv.org/html/2609.00054#bib.bib53)and[Fokou et al\. \(2025\)](https://arxiv.org/html/2609.00054#bib.bib34), the process of pattern extraction requires to remove some redundant links\. For example, remedy conceptcr\_0\(see Fig\.[3](https://arxiv.org/html/2609.00054#S2.F3)\) has eight relational attributes pointing atcp\_1,cp\_4,cp\_5,cp\_6,cp\_7,cp\_9,cp\_10andcp\_11\. Plant conceptcp\_11\(top concept, see Fig\.[1](https://arxiv.org/html/2609.00054#S2.F1)\) brings no information as well ascp\_7\. Plant conceptscp\_6,cp\_9andcp\_10introduce attributes that are inherited by lower concepts; besides,cp\_9introduces an object that is not linked to remedies ofcr\_0\(remedy0andremedy2\)\. Finally, in the graph built on this lattice family \(see Fig\.[4](https://arxiv.org/html/2609.00054#S2.F4)\), only conceptscp\_1\(denotedplants\_1\),cp\_4\(plants\_4\), andcp\_5\(plants\_5\) are represented as linked tocr\_0\(remedies\_0\)\.

## 3Relational Concept Analysis based on AOC\-posets

A variant of RCA based on AOC\-posets –called RCA\-AOC– was introduced by[Dolques et al\. \(2013b\)](https://arxiv.org/html/2609.00054#bib.bib28)\. We here explain its principles relying on the same relational context family\. The AOC\-poset associated with a concept lattice is its suborder restricted to concepts that introduce at least one object or one attribute\. The AOC\-posets built on the Plants\-Remedies relational context family are shown in Fig\.[5](https://arxiv.org/html/2609.00054#S3.F5)\. Three concepts have been removed from the Plant concept lattice of Fig\.[1](https://arxiv.org/html/2609.00054#S2.F1):cp\_11\(top\),cp\_0\(bottom\), andcp\_7\(internal\), leading to the Plant AOC\-poset on the right\-hand side of Fig\.[5](https://arxiv.org/html/2609.00054#S3.F5)\(note that the concepts have been renumbered between Fig\.[1](https://arxiv.org/html/2609.00054#S2.F1)and Fig\.[5](https://arxiv.org/html/2609.00054#S3.F5)\)\.

Figure 5:AOC\-posets of Remedies \(left\) and Plants \(right\)\.Now, since we rely on AOC\-posets, the existential scaling operator is defined slightly differently from classical RCA\([Rouane\-Hacène et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib60)\)\. While the scaling operation on RCA relies on the fact that all the concepts of the concept lattice have been created \(see Definitions[6](https://arxiv.org/html/2609.00054#Thmdefinition6)and[7](https://arxiv.org/html/2609.00054#Thmdefinition7)\), we acknowledge here that only a subset of the concept lattice, such as an AOC\-poset, can be considered to build relational attributes\. Then the previous definitions of scaling operators are unchanged except that𝒞tr\{\\cal C\}\_\{t\_\{r\}\}is any set of concepts, not necessarily the entire set of concepts of the concept lattice\.

Table[4](https://arxiv.org/html/2609.00054#S3.T4)shows𝕊∃​\(𝒦𝑅𝑒𝑚𝑒𝑑𝑖𝑒𝑠,r𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠,𝒞𝑃𝑙𝑎𝑛𝑡𝑠\)\\mathbb\{S\}\_\{\\exists\}\(\\mathcal\{K\}\_\{\\mathit\{Remedies\}\},r\_\{\\mathit\{contains\}\},\{\\cal C\}\_\{\\mathit\{Plants\}\}\)where𝒞P​l​a​n​t​s\{\\cal C\}\_\{Plants\}is the set of concepts of the AOC\-poset of the right\-hand side of Fig\.[5](https://arxiv.org/html/2609.00054#S3.F5)\.I\+I^\{\+\}contains\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹CLOSE,\(\\mathtt\{remedy3\},OPEN∃𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠⁡\(𝚌𝚙​\_​𝟷\)\)\\exists\\penalty\\ \{\\mathit\{contains\}\}\(\\mathtt\{cp\\\_1\}\)\),\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹CLOSE,\(\\mathtt\{remedy3\},OPEN∃𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠⁡\(𝚌𝚙​\_​𝟸\)\)\\exists\\penalty\\ \{\\mathit\{contains\}\}\(\\mathtt\{cp\\\_2\}\)\),\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹CLOSE,\(\\mathtt\{remedy3\},OPEN∃𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠⁡\(𝚌𝚙​\_​𝟻\)\)\\exists\\penalty\\ \{\\mathit\{contains\}\}\(\\mathtt\{cp\\\_5\}\)\),\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹CLOSE,\(\\mathtt\{remedy3\},OPEN∃𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠⁡\(𝚌𝚙​\_​𝟼\)\)\\exists\\penalty\\ \{\\mathit\{contains\}\}\(\\mathtt\{cp\\\_6\}\)\)and\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹,∃𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠⁡\(𝚌𝚙​\_​𝟾\)\)\(\\mathtt\{remedy3\},\\exists\\penalty\\ \{\\mathit\{contains\}\}\(\\mathtt\{cp\\\_8\}\)\)becausec​o​n​t​a​i​n​s\{contains\}\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹\)=\{c​a​m​o​m​i​l​e,g​a​r​l​i​c,b​o​r​a​g​e\}\(\\mathtt\{remedy3\}\)=\\\{camomile,garlic,borage\\\}and each of the pointed concepts contains at least one of those plants in its extent\.

Table 4:Existential Scaling of Formal Context of Remedies with AOC\-poset of Plants𝕊∃​\(KR​e​m​e​d​i​e​s,r𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠,𝒞P​l​a​n​t​s\)\\mathbb\{S\}\_\{\\exists\}\(K\_\{Remedies\},r\_\{\\mathit\{contains\}\},\{\\cal C\}\_\{Plants\}\)\.∃r⁡\(cp\_i\)\\exists r\(\\texttt\{cp\\\_i\}\)abbreviates∃r𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠​\(cp\_i\)\\exists\{r\_\{\\mathit\{contains\}\}\}\(\\texttt\{cp\\\_i\}\)\.In the case of the strict universal scaling operator,I\+=\{\(o,∃∀r\(C\)\)\|o∈G,C∈𝒞tr,r\(o\)⊆Extent\(C\)andr\(o\)≠∅\}I^\{\+\}=\\\{\(o,\\exists\\forall r\(C\)\)\|o\\in G,C\\in\{\\cal C\}\_\{t\_\{r\}\},r\(o\)\\subseteq Extent\(C\)\\penalty\\ and\\penalty\\ r\(o\)\\neq\\emptyset\\\}\. Table[5](https://arxiv.org/html/2609.00054#S3.T5)shows𝕊∃∀​\(𝒦R​e​m​e​d​i​e​sCLOSE,\\mathbb\{S\}\_\{\\exists\\forall\}\(\\mathcal\{K\}\_\{Remedies\},𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠,\\mathit\{contains\},OPEN𝒞𝑃𝑙𝑎𝑛𝑡𝑠\)\{\\cal C\}\_\{\\mathit\{Plants\}\}\)\. In this case,I\+I^\{\+\}contains e\.g\.\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹CLOSE,\(\\mathtt\{remedy3\},OPEN∃∀⁡𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠⁡\(𝚌𝚙​\_​𝟼\)\)\\exists\\forall\\penalty\\ \{\\mathit\{contains\}\}\(\\mathtt\{cp\\\_6\}\)\)because all plants found inremedy3belong to the extent of this concept\. ButI\+I^\{\+\}does not contain\(𝚛𝚎𝚖𝚎𝚍𝚢𝟹CLOSE,\(\\mathtt\{remedy3\},OPEN∃∀⁡𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠⁡\(𝚌𝚙​\_​𝟸\)\)\\exists\\forall\\penalty\\ \{\\mathit\{contains\}\}\(\\mathtt\{cp\\\_2\}\)\)becauseremedy3contains plants \(e\.g\.camomile\) that are not in the extent of this concept \(reduced togarlic\)\.

Table 5:Strict universal scaling of formal context of Remedies with AOC\-poset of Plants𝕊∃∀​\(KR​e​m​e​d​i​e​s,r𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠,𝒞P​l​a​n​t​s\)\\mathbb\{S\}\_\{\\exists\\forall\}\(K\_\{Remedies\},r\_\{\\mathit\{contains\}\},\{\\cal C\}\_\{Plants\}\)\.∃∀⁡r⁡\(cp\_i\)\\exists\\forall r\(\\texttt\{cp\\\_i\}\)abbreviates∃∀⁡r𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠​\(cp\_i\)\\exists\\forall\{r\_\{\\mathit\{contains\}\}\}\(\\texttt\{cp\\\_i\}\)\.remedies∃∀\\exists\\forallr\(cp\_2\)

∃∀\\exists\\forallr\(cp\_0\)

∃∀\\exists\\forallr\(cp\_4\)

∃∀\\exists\\forallr\(cp\_3\)

∃∀\\exists\\forallr\(cp\_5\)

∃∀\\exists\\forallr\(cp\_7\)

∃∀\\exists\\forallr\(cp\_8\)

∃∀\\exists\\forallr\(cp\_1\)

∃∀\\exists\\forallr\(cp\_6\)

remedy0×\\times×\\timesremedy1×\\timesremedy2×\\timesremedy3×\\times×\\timesFigure 6:AOC\-poset for theKR​e​m​e​d​i​e​sK\_\{Remedies\}formal context extended with the existential scaling on𝒞P​l​a​n​t​s\{\\cal C\}\_\{Plants\}\.As for RCA, the whole construction process consists in building a \(possibly infinite\) sequence of contexts and AOC\-posets associated with\(𝐊,𝐑\)\(\{\\mathbf\{K\}\},\{\\mathbf\{R\}\}\)andρ\\rho\. The difference here with classical RCA is that stepp\+1p\+1relies on the original context𝐊\{\\mathbf\{K\}\}and the concept set of AOC\-posets𝐂p\{\\mathbf\{C\}\}^\{p\}to build𝐊p\+1\{\\mathbf\{K\}\}^\{p\+1\}, rather than on𝐊p\{\\mathbf\{K\}\}^\{p\}and the set of concepts from lattices of steppp\(this is discussed in Section[5\.4](https://arxiv.org/html/2609.00054#S5.SS4)\)\.

In the current simple example, the following steps do not produce any new concept \(a fixpoint is reached\)\. More complex datasets may include cycles between objects\. This can lead to convergence problems in the RCA\-AOC process, as discussed in the following section\.

##### RCA\-AOC output as graphs

Obviously, patterns built from AOC\-posets are included in the corresponding patterns built from RCA lattices\. Furthermore, since building patterns requires removing redundancies, i\.e\. concepts that bring no information, building patterns based on AOC\-poset is more straightforward than based on RCA lattices and should lead to the same result\. For example, Remedy AO\-concept222An AO\-concept is a concept introducing at least one attribute or object, i\.e\. it belongs to the AOC\-poset\.cr\_0in Fig\.[6](https://arxiv.org/html/2609.00054#S3.F6)corresponds to conceptcr\_0in Fig\.[3](https://arxiv.org/html/2609.00054#S2.F3)\. It has only six relational attributes, pointing at Plant AO\-conceptscp\_0,cp\_3,cp\_4,cp\_5,cp\_7, andcp\_8\. Plant AO\-conceptscp\_5andcp\_8introduce only attributes that are inherited by lower AO\-concepts,cp\_7introduces an object that is not linked to remedies of AO\-conceptcr\_0\. In the final pattern onlycp\_0\(celery\),cp\_3\(woodruff,celery\) andcp\_4\(angelica,celery\) are linked to Remedy AO\-conceptcr\_0as for Remedy conceptcr\_0in Fig\.[4](https://arxiv.org/html/2609.00054#S2.F4)\.

## 4Convergence issues in RCA\-AOC

The convergence of an iterative process is the property of reaching a stable solution after a finite number of steps\. This notion is fundamental, as it enables the design of practical stopping criteria that can be effectively verified\. The stop condition for RCA and RCA\-AOC is the equivalence of concept\-posets \(concept lattice or AOC\-poset\) associated with the same initial context between two successive iterations\. Two concept\-posets are equivalent if the sets of their extents are equal, as the equivalence of two concepts is the equality of their extents\.

Convergence is ensured with the RCA specification from[Rouane\-Hacène et al\. \(2013\)](https://arxiv.org/html/2609.00054#bib.bib60)where the set of concepts used at each step for building relational attributes is the set of concepts of the whole lattice\. Besides, it makes it possible to interpret the last built lattices using only the last step lattices, while forgetting the lattices of the previous steps\. By interpretation, we mean that every concept occurring in the expression of a relational attribute at the last step is present in one of the lattices obtained at this step\. As a consequence, we can interpret these attributes recursively, as done by[Gutierrez et al\. \(2025\)](https://arxiv.org/html/2609.00054#bib.bib41), or extract a graph representation, as in\([Nica et al\., 2020a](https://arxiv.org/html/2609.00054#bib.bib52);[Nica et al\., 2020b](https://arxiv.org/html/2609.00054#bib.bib53)\), without encountering any concept that is not defined at this step\. Unfortunately, RCA\-AOC may not converge in the general case\. In this section, we illustrate this divergence through several examples based on the two scaling operators∃\\existsand∃∀\\exists\\forall\. The existential scaling operator is the most commonly used in practice, and the strict universal scaling operator is a natural alternative for discovering more restricted concepts\. The first example \(Sect\.[4\.1](https://arxiv.org/html/2609.00054#S4.SS1)\) is a concrete situation that may arise when applying RCA\-AOC in the software engineering domain, more specifically when using it as a tool to support conceptual model refactoring\. The next two examples \(Sect\.[4\.2](https://arxiv.org/html/2609.00054#S4.SS2)\) are small formal cases that highlight typical divergence patterns for∃\\existsand∃∀\\exists\\foralloperators respectively\.

### 4\.1A concrete divergent example

Divergence may arise in various concrete situations\. In this section, we consider an application of FCA and RCA to software engineering\. This application, developed over the years\([Godin and Mili, 1993](https://arxiv.org/html/2609.00054#bib.bib37);[Dao et al\., 2004](https://arxiv.org/html/2609.00054#bib.bib23);[Huchard et al\., 2007](https://arxiv.org/html/2609.00054#bib.bib46);[Miralles et al\., 2015](https://arxiv.org/html/2609.00054#bib.bib50);[Guenoune et al\., 2025](https://arxiv.org/html/2609.00054#bib.bib40)\), aims to improve the abstraction level of object\-oriented code or conceptual models \(e\.g\. database schemas, ER diagrams, or UML class models\) by adding new elements \(classes, attributes, operations, etc\.\) that factor out common information and represent more general notions\. The example we focus on is taken from the UML world\. It is deliberately simple and only aims to illustrate concretely the possibility of divergence in this process\.

A UML class model is primarily composed of classes that represent notions \(concepts\) relevant to the software being developed\. For example, in software dedicated to the banking domain, one might introduce a classBankAccount\. Additional information is attached to these classes, such as properties \(also called attributes\) and operations \(also called methods\)\. For instance, the classBankAccountcan be enriched with a propertyaccountNumberand an operationwithdraw\. These elements are themselves described by further information, such as the type of a property \(e\.g\.String\) or the parameters of an operation \(e\.g\.amountof typeReal\)\.

The UML metamodel, as defined by the OMG\([Object Management Group, 2017](https://arxiv.org/html/2609.00054#bib.bib54)\), specifies all the entities that may appear in a UML model, and in particular in a class model\. We here rely on an excerpt of this metamodel, sketched in Fig\.[7](https://arxiv.org/html/2609.00054#S4.F7)\. A class owns operations and properties\. Classes and properties are described by their name\. In addition, properties have a type\. An operation owns parameters\. Each parameter has a name, a direction, whereParameterDirectionKindisin, inout, out, or return, and a type\. This excerpt deliberately omits some meta\-classes and meta\-attributes of the UML metamodel — notably the name meta\-attribute of Operation — so that the two operations are indistinguishable at first and are initially grouped into a single concept\.

Figure 7:Excerpt of the OMG UML metamodel\. Arrows represent UML associations\. Since we are describing a metamodel, the various elements \(classes, associations, etc\.\) are referred to as meta\-elements\.
Figure 8:Top: A simple UML model in the Bank domain\. Bottom: a lexical resource\. Arrows represent a hyponymy relation, pointing from the hyponym to the hypernym\.
An example of a succinct UML class model in the banking domain is shown at the top of Fig\.[8](https://arxiv.org/html/2609.00054#S4.F8)\. This class diagram \(Bank model\) comprises four classes,Ledger,BankAccount,CustomerandTransaction\.BankAccounthas been described previously\.Ledgerowns the propertyfiscalPeriodand the operationreconcileEntrieswhich returns a Boolean\.Customeris simply described by the propertyaddress, whileTransactionhas the propertytransID\.

To analyze the UML banking class model, for example for refactoring purposes, we need a description language for UML models, that is a UML metamodel\. Based on this metamodel, we can encode the banking class model as a Relational Context Family\. There are various ways of proceeding; a simple one consists in associating: \(1\) a formal context to each meta\-class of the UML metamodel \(i\.e\.Class,Operation,Parameter,Property\); \(2\) a relational context to each meta\-association \(i\.e\.ownedOperation,class,ownedAttribute,ownedParameter\)\. This is reflected respectively in the structure of Tables[6](https://arxiv.org/html/2609.00054#S4.T6)and[7](https://arxiv.org/html/2609.00054#S4.T7)\.

Table 6:Formal Contexts of the Bank UML fragmentClass

Operation

Parameter

Property

Table 7:Relational Contexts of the Bank UML fragment\. For the sake of simplicity,ownedAttributeis the only relational context for which we consider an inverse \(class\) in this example\.ownedOperation

ownedParameter

ownedAttribute

class

The formal and relational contexts are then populated with the elements of the UML model, here the simple Bank model\. For this step, the UML class model is viewed as an instantiation of the metamodel of Fig\.[7](https://arxiv.org/html/2609.00054#S4.F7)\(the UML instance diagram is then used\)\. This instantiation is presented in Fig\.[9](https://arxiv.org/html/2609.00054#S4.F9)\. It indicates for example thatLedgeris aClass,fiscalPeriodis aProperty,withdrawis anOperationwhich hasamountas its inputParameter\. It is common to also consider additional information, e\.g\. information taken from a lexical resource that can be added as attributes of the contexts\. For example, the bottom of Fig\.[8](https://arxiv.org/html/2609.00054#S4.F8)shows a possible lexical resource on attribute names\. Arrows represent a hyponymy relation\.

Figure 9:The UML model of Fig\.[8](https://arxiv.org/html/2609.00054#S4.F8)presented as an instantiation of the UML metamodel of Fig\.[7](https://arxiv.org/html/2609.00054#S4.F7)\.
The RCA\-AOC process can now be started\. In this application, the∃\\existsoperator is used, since the refactoring principle consists in adding a new superclass to classes that share at least one element \(attribute or operation\) appearing in the extent of a concept\.

Figure 10:Bank example \- AOC\-posets at step 0\. In the paper,C\_Operation\_imay be shortened toco\_i,C\_Parameter\_imay be shortened tocpa\_i,C\_Class\_imay be shortened tocc\_i, andC\_Property\_imay be shortened tocpr\_i\. For each concept, I, E, and Sta denote the intent cardinality, extent cardinality, and stability metric in decimal comma notation, respectively\.At step 0 of the RCA\-AOC process \(Fig\.[10](https://arxiv.org/html/2609.00054#S4.F10)\), both operations are grouped into the operation conceptco\_8, as they have no attributes and are indistinguishable at first\. Parameters are separated into distinct conceptscpa\_9andcpa\_10\. Similarly classes are distributed in conceptscc\_4tocc\_7\. The property conceptcpr\_15groups properties whose type isString\. The property conceptcpr\_16groups properties whose name is a hyponym ofadminId\.

Figure 11:Bank example \- AOC\-posets at step 1\.At step 1 \(Fig\.[11](https://arxiv.org/html/2609.00054#S4.F11)\), several relational attributes are introduced\. Firstly∃\\exists\_ownedAttribute\(C\_Property\_15\), aka∃\\exists\_ownedAttribute\(type=String\)if we replace the concept by its intent, is shared byCustomer,LedgerandBankAccountand factored out in conceptcc\_18\. Secondly,∃\\exists\_ownedAttribute\(cpr\_16\), aka∃\\exists\_ownedAttribute\(name=adminId\)if we replace the concept by its intent, is shared byTransaction,LedgerandBankAccountand factored out in conceptcc\_19\. Lastly,∃\\exists\_ownedOperation\(co\_8\)is shared byLedgerandBankAccountand factored out in conceptcc\_17\. Let us note that this last relational attribute refers to a conceptco\_8absent at this step 1 by construction of the relational attributes that are based on concepts of the previous step, i\.e\. step 0\. Each bottom property concept \(cpr\_11tocpr\_14\) is completed with a relational attribute pointing to the class that owns the property, e\.g\.∃\\exists\_class\(cc\_6\)is added asaddressis owned byCustomer, belonging tocc\_6extent\.

Figure 12:Bank example \- Class and Property AOC\-posets at step 2 \(Operation and Parameter AOC\-posets are unchanged\)\.At step 2 \(Fig\.[12](https://arxiv.org/html/2609.00054#S4.F12)\),cc\_17, which existed at step 1, no longer exists, asco\_8, a concept that existed at step 0, no longer existed at step 1\. But ascc\_17was present at step 1, it is used to buildcpr\_22, to factor out∃\\exists\_class\(cc\_17\)which is shared byfiscalPeriodandaccountNumber\.

Figure 13:Bank example \- Class and Property AOC\-posets at step 3 \(Operation and Parameter AOC\-posets are unchanged\)\.At step 3 \(Fig\.[13](https://arxiv.org/html/2609.00054#S4.F13)\),cpr\_22of step 2 is used to buildcc\_23, to factor out∃\\exists\_ownedAttribute\(cpr\_22\), which is shared byLedgerandBankAccount\. Butcpr\_22disappears at this step, since conceptcc\_17does not exist at step 2\.

Class and Property AOC\-posets from step 4 are equivalent to those from step 2 because they refer to concepts of step 3 and 1 respectively, that are equivalent \(up to a renaming of the concepts\)\. From step 1 to 2, the same concepts have disappeared as from step 3 to 4\. These configurations will therefore alternate indefinitely\.

The AOC\-posets of step 1 and 2 can be used to suggest an evolution of the banking model of Fig\.[8](https://arxiv.org/html/2609.00054#S4.F8)\.cpr\_22\(step 2\) suggests introducing a new UML propertyadminId:Stringthat can be specialized by UML propertiesfiscalPeriodandaccountNumber, using UML constraint syntax\{redefines adminId:String\}\.cc\_17\(step 1\) suggests a UML class that a software engineer may callFinancialStructure, on the basis of the factored out propertyadminId:Stringand the names of the subclassesLedgerandBankAccount\.cc\_19\(step 1\) suggests a UML classAdminAssetto factor outadminId\.

The refactored UML class model is shown in Fig\.[14](https://arxiv.org/html/2609.00054#S4.F14)\. The names of the new superclasses have to be found by the software engineer, possibly assisted by a lexical resource or a generative AI agent\([Guenoune et al\., 2025](https://arxiv.org/html/2609.00054#bib.bib40)\)\.

Figure 14:Refactoring suggestions for the Bank model\. The two new superclasses factor out the notion of owning an administrative identification \(inAdminAsset\), and the fact that this identification is a String \(inFinancialStructure\), and group coherent classes\.
### 4\.2Diverging examples

In this section, we present two small formal examples highlighting the mechanism of divergence for two quantifiers\.

#### 4\.2\.1Existential scaling

This example reduces the UML divergence example to its essence\. The RCF is presented in Table[8](https://arxiv.org/html/2609.00054#S4.T8), and the first steps of the process are illustrated by Table[9](https://arxiv.org/html/2609.00054#S4.T9)with all the created AOC\-posets\.

Table 8:RCF for the counterexample illustrating divergence using the existential scaling operator \(∃\\exists\) on each relation\.Table 9:AOC\-posets obtained from applying RCA\-AOC on the RCF from Table[8](https://arxiv.org/html/2609.00054#S4.T8)using the existential scaling operator \(∃\\exists\) on each relation\. For each step, AOC\-posets come, from left to right, fromK1K\_\{1\},K2K\_\{2\},K3K\_\{3\}andK4K\_\{4\}\.Let us look at the AOC\-posets built during the process and shown in Table[9](https://arxiv.org/html/2609.00054#S4.T9)\. To help the reader, a concept name contains the context from which the concept is built \(e\.g\. C\_K1\_0 is the first concept built from theK1K\_\{1\}context\) and the same name is reused for the same concept in the following steps\. In step 0, seven concepts are created, one for contextK1K\_\{1\}, and two in the other contexts\. In step 1, C\_K4\_2 is created depending on C\_K1\_0 from step 0 and contains no object in its simplified extent\. So as C\_K1\_0 is removed in step 1, it leads to the removal of C\_K4\_2 in step 2 with no creation of new concepts in this AOC\-poset\. However C\_K3\_2 has been created in step 2 because of the presence of C\_K4\_2 in step 1, and has an empty simplified extent\. In step 3, C\_K3\_2 is thus removed, and no new concept is created in this AOC\-poset\. In the same step, C\_K4\_2 reappears because of C\_K3\_2: we have an inter\-dependency between two concepts in a way that they will appear alternately\. Step 3 is equivalent to step 1, step 4 will be equivalent to step 2, step 5 to step 3, and so on\. This leads to an infinite loop\.

#### 4\.2\.2Strict Universal scaling

Table[10](https://arxiv.org/html/2609.00054#S4.T10)presents an RCF which, if we use the strict universal operator for every relation, makes the RCA\-AOC process loop infinitely\. The first steps are detailed in Table[11](https://arxiv.org/html/2609.00054#S4.T11)by showing the AOC\-posets obtained\. The divergence of the process appears at step 4 as we obtain AOC\-posets equivalent to those of step 0, which means that step 5 will be equivalent to step 1, etc\.

Table 10:RCF for the counterexample illustrating divergence using the strict universal scaling operator \(∃∀\\exists\\forall\) on each relation\.Table 11:AOC\-posets obtained from applying RCA\-AOC on the relational context family from Table[10](https://arxiv.org/html/2609.00054#S4.T10)using the strict universal scaling operator \(∃∀\\exists\\forall\) on each relation\. For each step, AOC\-posets come, from left to right, fromK1K\_\{1\},K2K\_\{2\}andK3K\_\{3\}\.Let us look at the AOC\-posets built during the process and shown in Table[11](https://arxiv.org/html/2609.00054#S4.T11)\. Concepts are named as previously\. The existence of C\_K3\_2 and C\_K3\_3 \(which appear at step 1\) depends on the existence of C\_K2\_1 and C\_K1\_1\. If C\_K2\_1 and C\_K1\_1 are not present at the previous step then C\_K3\_2 and C\_K3\_3 cannot be created as we see at steps 3 and 4\. Instead their extents \(composed respectively of o6 and o5\) are contained in the extent of C\_K3\_1, that cannot appear simultaneously with C\_K3\_2 and C\_K3\_3 in an AOC\-poset\.

On the other hand, C\_K1\_2, C\_K1\_3, C\_K2\_2 and C\_K2\_3 \(steps 2 and 3\) depend on the existence of C\_K3\_2 and C\_K3\_3 at the previous step\. When C\_K1\_2, C\_K1\_3, C\_K2\_2 and C\_K2\_3 appear, C\_K1\_1 and C\_K2\_1 cannot appear at the same step\. Those inter\-dependencies lead to the divergence of the process\.

These two counterexamples show that convergence in RCA\-AOC is not always ensured\. We present in the following some conditions to ensure it\.

## 5Approaches to Ensuring RCA\-AOC Convergence

The lack of convergence guarantee is a threat that could discourage users from adopting RCA\-AOC\. Indeed, the occurrence of a divergence case would make the result far more difficult to interpret\. To resolve this threat, and in order to make our method robust we need to find ways to guarantee convergence with as little impact as possible on data\. In this section, we discuss different approaches to ensuring convergence, either by requiring appropriate constraints on the dataset or by introducing a convergent variant of the RCA\-AOC process\.

### 5\.1Preliminaries

We introduce here some useful definitions\. We first define basic notions regarding contexts and concept\-posets\. We then define the notion of identity of relational attributes, and the convergence for RCA\-AOC, that relies on the existence of a fixpoint for each concept\-poset of the relational family\. Finally we define the notion of dependency graph for an RCF and introduce the notion of identified objects\.

###### Definition 9\(Concept and Concept\-poset equivalences\)\.

LetC1C\_\{1\}andC2C\_\{2\}be two concepts from two different concept\-posets𝒜1\\mathcal\{A\}\_\{1\}and𝒜2\\mathcal\{A\}\_\{2\}\.C1C\_\{1\}andC2C\_\{2\}are equivalent, denotedC1∼C2C\_\{1\}\\sim C\_\{2\}, iff𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C1\)=𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C2\)\\mathit\{Extent\}\(C\_\{1\}\)=\\mathit\{Extent\}\(C\_\{2\}\)\.𝒜1\\mathcal\{A\}\_\{1\}and𝒜2\\mathcal\{A\}\_\{2\}are equivalent iffE​x​t𝒜1=E​x​t𝒜2Ext\_\{\\mathcal\{A\}\_\{1\}\}=Ext\_\{\\mathcal\{A\}\_\{2\}\}whereE​x​t𝒜1=⋃C∈𝒜1\{𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)\}Ext\_\{\\mathcal\{A\}\_\{1\}\}=\\bigcup\_\{C\\in\\mathcal\{A\}\_\{1\}\}\\\{\\mathit\{Extent\(C\)\}\\\}andE​x​t𝒜2=⋃C∈𝒜2\{𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)\}Ext\_\{\\mathcal\{A\}\_\{2\}\}=\\bigcup\_\{C\\in\\mathcal\{A\}\_\{2\}\}\\\{\\mathit\{Extent\(C\)\}\\\}\. We denote it𝒜1≡𝒜2\\mathcal\{A\}\_\{1\}\\equiv\\mathcal\{A\}\_\{2\}\.

Example: concept\-posets from steps 3 and 4 in Table[11](https://arxiv.org/html/2609.00054#S4.T11)are such thatE​x​t𝒜13=\{\{o1\},\{o2\},∅\}Ext\_\{\\mathcal\{A\}\_\{1\}^\{3\}\}=\\\{\\\{o\_\{1\}\\\},\\\{o\_\{2\}\\\},\\emptyset\\\},E​x​t𝒜23=\{\{o3\},\{o4\},∅\}Ext\_\{\\mathcal\{A\}\_\{2\}^\{3\}\}=\\\{\\\{o\_\{3\}\\\},\\\{o\_\{4\}\\\},\\emptyset\\\},E​x​t𝒜33=\{\{o5,o6\},∅\}Ext\_\{\\mathcal\{A\}\_\{3\}^\{3\}\}=\\\{\\\{o\_\{5\},o\_\{6\}\\\},\\emptyset\\\}whileE​x​t𝒜14=\{\{o1,o2\},∅\}Ext\_\{\\mathcal\{A\}\_\{1\}^\{4\}\}=\\\{\\\{o\_\{1\},o\_\{2\}\\\},\\emptyset\\\},E​x​t𝒜24=\{\{o3,o4\},∅\}Ext\_\{\\mathcal\{A\}\_\{2\}^\{4\}\}=\\\{\\\{o\_\{3\},o\_\{4\}\\\},\\emptyset\\\}, andE​x​t𝒜34Ext\_\{\\mathcal\{A\}\_\{3\}^\{4\}\}=E​x​t𝒜33=Ext\_\{\\mathcal\{A\}\_\{3\}^\{3\}\}\.

###### Definition 10\(Identity of relational attributes\)\.

The identity of a relational attributeρ⁡\(r\)​r​\(C\)\\rho\(r\)\\,r\(C\)is defined as the triple\(ρ⁡\(r\),r,𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)\)\(\\rho\(r\),r,\\mathit\{Extent\}\(C\)\)\. Then, two relational attributes built, possibly at different steps, with the same quantifier and the same relation, and on equivalent concepts, are the same attribute\.

The incidence of a relational attribute is determined at its creation and is never recomputed afterwards; it only depends on this identifying triple\. E\.g\. Table[11](https://arxiv.org/html/2609.00054#S4.T11), the relational attribute∃∀\\exists\\forallR2\(C\_K3\_1\) is the same in C\_K2\_1 at step 1 and step 4\. Note that, in this example, equivalent concepts occurring at different steps have the same identifier\.

We now introduce two definitions for context growing, the first one based on context inclusion, the second one based on the inclusion of sets of extents in the associated posets\.

###### Definition 11\(Context inclusion and monotonic growth\)\.

Let𝒦=\(G,M,I\)\\mathcal\{K\}=\(G,M,I\)and𝒦′=\(G,M′,I′\)\\mathcal\{K\}^\{\\prime\}=\(G,M^\{\\prime\},I^\{\\prime\}\)be two contexts\.𝒦⊆𝒦′\\mathcal\{K\}\\subseteq\\mathcal\{K\}^\{\\prime\}iffM⊆M′M\\subseteq M^\{\\prime\}andI=I′∩\(G×M\)I=I^\{\\prime\}\\cap\(G\\times M\)\. Furthermore, let\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)be an RCF, and𝒦i\{\\cal K\}\_\{i\}a context of𝐊\\mathbf\{K\}\. Let𝒦i0,𝒦i1,…​𝒦in,…\\mathcal\{K\}\_\{i\}^\{0\},\\mathcal\{K\}\_\{i\}^\{1\},\.\.\.\\mathcal\{K\}\_\{i\}^\{n\},\.\.\.be the succession of extended formal contexts of𝒦i\\mathcal\{K\}\_\{i\}in the RCA\-AOC process\. The context𝒦i\\mathcal\{K\}\_\{i\}is growing monotonically between stepsjjandkkiff∀l∈\[j\.\.k\[\\forall l\\in\[j\.\.k\[𝒦il⊆𝒦il\+1\\mathcal\{K\}\_\{i\}^\{l\}\\subseteq\\mathcal\{K\}\_\{i\}^\{l\+1\}\. The context is said to grow monotonically from stepjjwhenk=∞k=\\infty\.

###### Definition 12\(Context monotonic poset\-growth\)\.

Let\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)be an RCF, and𝒦i\{\\cal K\}\_\{i\}a context of𝐊\\mathbf\{K\}\. Let𝒦i0,𝒦i1,…​𝒦in,…\\mathcal\{K\}\_\{i\}^\{0\},\\mathcal\{K\}\_\{i\}^\{1\},\.\.\.\\mathcal\{K\}\_\{i\}^\{n\},\.\.\.be the succession of extended formal contexts of𝒦i\\mathcal\{K\}\_\{i\}in the RCA\-AOC process, and𝒜i0,𝒜i1,…​𝒜in,…\\mathcal\{A\}\_\{i\}^\{0\},\\mathcal\{A\}\_\{i\}^\{1\},\.\.\.\\mathcal\{A\}\_\{i\}^\{n\},\.\.\.the corresponding concept\-posets\. The context𝒦i\\mathcal\{K\}\_\{i\}is poset\-growing monotonically between stepsjjandkkiff∀l∈\[j\.\.k\[\\forall l\\in\[j\.\.k\[,E​x​t𝒜il⊆E​x​t𝒜il\+1Ext\_\{\\mathcal\{A\}\_\{i\}^\{l\}\}\\subseteq Ext\_\{\\mathcal\{A\}\_\{i\}^\{l\+1\}\}\(also denoted𝒜il⊆𝒜il\+1\\mathcal\{A\}\_\{i\}^\{l\}\\subseteq\\mathcal\{A\}\_\{i\}^\{l\+1\}\)\. The context is said to poset\-grow monotonically from stepjjwhenk=∞k=\\infty\.

In the RCA\-AOC process, each concept\-poset corresponds to a possibly extended object\-attribute context\. A concept\-poset fixpoint refers to the final concept\-poset generated from this context if it exists\. The verification of the fixpoint relies on the equivalence of successive concept\-posets\.

###### Definition 13\(Concept\-poset fixpoint\)\.

Let\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)be an RCF, and𝒦i\{\\cal K\}\_\{i\}a context of𝐊\\mathbf\{K\}\. Let𝒦i0,𝒦i1,…​𝒦in,…\\mathcal\{K\}\_\{i\}^\{0\},\\mathcal\{K\}\_\{i\}^\{1\},\.\.\.\\mathcal\{K\}\_\{i\}^\{n\},\.\.\.be the succession of extended formal contexts of𝒦i\\mathcal\{K\}\_\{i\}in the RCA\-AOC process, and𝒜i0,𝒜i1,…​𝒜in,…\\mathcal\{A\}\_\{i\}^\{0\},\\mathcal\{A\}\_\{i\}^\{1\},\.\.\.\\mathcal\{A\}\_\{i\}^\{n\},\.\.\.the corresponding concept\-posets\. If there existsppsuch that for allq≥pq\\geq p𝒜iq≡𝒜iq\+1\\mathcal\{A\}\_\{i\}^\{q\}\\equiv\\mathcal\{A\}\_\{i\}^\{q\+1\}, then the context𝒦i\{\\cal K\}\_\{i\}admits a concept\-poset fixpoint,𝒜ip\\mathcal\{A\}\_\{i\}^\{p\}, associated to the extended context𝒦ip\\mathcal\{K\}\_\{i\}^\{p\}of steppp\.

###### Definition 14\(Convergence of RCA\-AOC\)\.

The convergence of an application of RCA\-AOC on a relational context family\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)is determined by the existence of a fixpoint in the successively generated concept\-poset families, i\.e\. by the existence of a concept\-poset fixpoint for each original context𝒦i∈𝐊\{\\cal K\}\_\{i\}\\in\\mathbf\{K\}\.

If all concept\-posets reach a fixpoint, then the process can be stopped in practice at stepnnsuch that the last concept\-poset reaches its own fixpoint, i\.e\.∀i,𝒜in≡𝒜in−1\\forall i,\\mathcal\{A\}\_\{i\}^\{n\}\\equiv\\mathcal\{A\}\_\{i\}^\{n\-1\}\.

###### Definition 15\(Dependency graph\)\.

The dependency graph of a relational context familyR​C​F=\(𝐊,𝐑\)RCF=\(\\mathbf\{K\},\\mathbf\{R\}\)is defined as the graph𝐆=\(𝐊,𝐄\)\\mathbf\{G\}=\(\\mathbf\{K\},\\mathbf\{E\}\), with𝐄=\{\(𝒦sr,𝒦tr\)∣r∈𝐑\}\\mathbf\{E\}=\\\{\(\\mathcal\{K\}\_\{s\_\{r\}\},\\mathcal\{K\}\_\{t\_\{r\}\}\)\\mid r\\in\\mathbf\{R\}\\\}\.

The dependency graph shows which contexts have to be considered for the analysis of a given object\-object context\. Figure[15](https://arxiv.org/html/2609.00054#S5.F15)shows the dependency graph for the RCF of Table[10](https://arxiv.org/html/2609.00054#S4.T10)\.

Figure 15:Dependency graph for RCF in Table[10](https://arxiv.org/html/2609.00054#S4.T10)\.###### Definition 16\(Identified object\)\.

Let𝒦=\(G,M,I\)\{\\cal K\}=\(G,M,I\)be an initial object\-attribute context from an RCF\. An objecto∈Go\\in Gsuch that\{o\}′′\\\{o\\\}^\{\\prime\\prime\}=\{o\}\\\{o\\\}is called an identified object\.

###### Definition 17\(Context of identified objects\)\.

A context of identified objects is an object\-attribute context𝒦𝑖𝑑=\(G,M,I\)\\mathcal\{K\_\{\\mathit\{id\}\}\}=\(G,M,I\)where∀o∈G,\{o\}′′=\{o\}\\forall o\\in G,\\\{o\\\}^\{\\prime\\prime\}=\\\{o\\\}\. As a consequence, concepts of𝒜𝑖𝑑\\mathcal\{A\_\{\\mathit\{id\}\}\}with a non\-empty simplified extent have an extent of size 1\.

Based on these definitions, we first examine different configurations of the dependency graph and convergence conditions \(Sect\.[5\.2](https://arxiv.org/html/2609.00054#S5.SS2)\)\. Then, we propose a process that ensures convergence \(Sect\.[5\.3](https://arxiv.org/html/2609.00054#S5.SS3)\)\. Finally, we introduce a variant of RCA\-AOC \(Sect\.[5\.4](https://arxiv.org/html/2609.00054#S5.SS4)\)\.

### 5\.2Convergence conditions

We examine here the conditions of convergence of RCA\-AOC, based on the structure of the dependency graph and some properties of the contexts\. A first lemma states that if a context grows monotonically, then this context reaches a fixpoint\.

###### Lemma 1\.

Let\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)be an RCF, and𝒦i\\mathcal\{K\}\_\{i\}a context of𝐊\\mathbf\{K\}\. Let𝒦i0,𝒦i1,…,𝒦in,\\mathcal\{K\}\_\{i\}^\{0\},\\mathcal\{K\}\_\{i\}^\{1\},\\ldots,\\mathcal\{K\}\_\{i\}^\{n\},…\\ldotsbe the succession of extended formal contexts of𝒦i\\mathcal\{K\}\_\{i\}in the RCA\-AOC process\. If there existsq∈ℕq\\in\\mathbb\{N\}from which𝒦i\\mathcal\{K\}\_\{i\}grows monotonically, then it admits a concept\-poset fixpoint\.

This follows directly from the fact that the maximal size of the extended context is bounded, and from a certain step it will therefore stop changing\. Then the associated concept\-poset will also be stable\.

The following theorem states that, if all the successors in𝐆\\mathbf\{G\}of an object\-attribute context have a concept\-poset fixpoint, then this context has a concept\-poset fixpoint too\.

###### Theorem 2\.

Let𝐆\\mathbf\{G\}be a dependency graph of a relational context family\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)\. Let𝒦i\\mathcal\{K\}\_\{i\}be a context of𝐊\\mathbf\{K\}\. We defineSi=\{𝒦j∈𝐊\|\(𝒦i,𝒦j\)S\_\{i\}=\\\{\\mathcal\{K\}\_\{j\}\\in\\mathbf\{K\}\|\(\\mathcal\{K\}\_\{i\},\\mathcal\{K\}\_\{j\}\)is an edge of𝐆\}\\mathbf\{G\}\\\}, the set of direct successors of𝒦i\\mathcal\{K\}\_\{i\}\. If all elements ofSiS\_\{i\}admit a concept\-poset fixpoint, then𝒦i\\mathcal\{K\}\_\{i\}also admits a concept\-poset fixpoint\.

This follows directly from the fact that if a context verifies the previous condition then its corresponding extended formal context will stop changing on the step following the one where all its successors in the dependency graph have reached their concept\-poset fixpoint333By Definition[10](https://arxiv.org/html/2609.00054#Thmdefinition10), the identity of a relational attribute only depends on the extent of the concept it refers to, thus the relational attributes do not change once their target concept\-posets are stable\.\.

Based on Theorem[2](https://arxiv.org/html/2609.00054#Thmtheorem2), the first sufficient property for the convergence results from the absence of circuits in the dependency graph\.

###### Corollary 3\.

If the dependency graph𝐆\\mathbf\{G\}of a relational context family is without circuit, then the RCA\-AOC process will converge\.

It is easy and fast to verify: all sink contexts reach their concept\-poset fixpoints at the first step since they do not depend on other contexts \(they have no successor\)\. Then, applying Theorem[2](https://arxiv.org/html/2609.00054#Thmtheorem2)recursively, we can propagate this result to the predecessors\.

However, in case of a dependency graph with circuits, the monotonic growth of contexts is not ensured, and contexts can lose attributes\. For example, in Table[11](https://arxiv.org/html/2609.00054#S4.T11), from step 3 to 4, contextK2K\_\{2\}loses 2 attributes \(pointing toC\_K3\_2andC\_K3\_3\), and is extended with one attribute \(pointing toC\_K3\_1\)\. Thus𝒦23⊈𝒦24\\mathcal\{K\}\_\{2\}^\{3\}\\not\\subseteq\\mathcal\{K\}\_\{2\}^\{4\}\. We need to add a constraint on a context to ensure its monotonic growth\. Theorem[5](https://arxiv.org/html/2609.00054#Thmtheorem5)states that adding an identifier to each object of a context can ensure the monotonic growth of the concept\-poset\. It relies on Lemma[4](https://arxiv.org/html/2609.00054#Thmtheorem4)\.

###### Lemma 4\.

Adding an attribute to a context of identified objects𝒦i​d\{\\mathcal\{K\}\_\{id\}\}cannot remove concepts in the corresponding concept\-poset𝒜i​d\{\\mathcal\{A\}\_\{id\}\}\(𝒜i​d\{\\mathcal\{A\}\_\{id\}\}grows\)\.

The proof of the lemma is given in Appendix[A](https://arxiv.org/html/2609.00054#A1)\. From this lemma we derive the proof of Theorem[5](https://arxiv.org/html/2609.00054#Thmtheorem5): if all successors of a context of identified objects poset\-grow monotonically, then at each step, this context will receive new attributes and its associated poset will grow\. Then monotonic growth of𝒜i​d\{\\mathcal\{A\}\_\{id\}\}is ensured\.

###### Theorem 5\.

Let\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)be an RCF,𝐆\\mathbf\{G\}its dependency graph, and𝒦i​d\{\\mathcal\{K\}\_\{id\}\}a context of identified objects in𝐊\\mathbf\{K\}\. LetSi​dS\_\{id\}be the set of𝒦i​d\{\\mathcal\{K\}\_\{id\}\}’s direct successors in𝐆\\mathbf\{G\}\. If there existsn∈ℕn\\in\\mathbb\{N\}from which all𝒦j∈Si​d\\mathcal\{K\}\_\{j\}\\in S\_\{id\}poset\-grow monotonically then for all stepsq≥n\+1q\\geq n\+1,𝒜i​dq⊆𝒜i​dq\+1\\mathcal\{A\}\_\{id\}^\{q\}\\subseteq\\mathcal\{A\}\_\{id\}^\{q\+1\}\. Thus,𝒦i​d\{\\mathcal\{K\}\_\{id\}\}admits a concept\-poset fixpoint\.

Based on Lemma[4](https://arxiv.org/html/2609.00054#Thmtheorem4)and Theorem[5](https://arxiv.org/html/2609.00054#Thmtheorem5)we show that convergence can be ensured in relational schemas with circuits if some contexts are contexts of identified objects\.

###### Corollary 6\(Convergence on circuits with identified objects\)\.

Let\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\)be an RCF and𝐒⊆𝐊\\mathbf\{S\}\\subseteq\\mathbf\{K\}a set of contexts, closed under the successor relation of the dependency graph \(i\.e\. every successor of a context of𝐒\\mathbf\{S\}is in𝐒\\mathbf\{S\}\), such that every context of𝐒\\mathbf\{S\}is either a sink or a context of identified objects\. Then every context of𝐒\\mathbf\{S\}poset\-grows monotonically and admits a concept\-poset fixpoint\.

###### Proof\.

We proceed by induction on the steps of RCA\-AOC\. At step00, contexts contain no relational attribute, hence between steps00and11every context of𝐒\\mathbf\{S\}only gains attributes\. Then for any context of identified objects𝒦i​d\\mathcal\{K\}\_\{id\}, the associated poset grows, hence𝒜i​d0⊆𝒜i​d1\\mathcal\{A\}^\{0\}\_\{id\}\\subseteq\\mathcal\{A\}^\{1\}\_\{id\}\(Lemma[4](https://arxiv.org/html/2609.00054#Thmtheorem4)\)\. Sinks are not extended:𝒜s0≡𝒜s1\\mathcal\{A\}^\{0\}\_\{s\}\\equiv\\mathcal\{A\}^\{1\}\_\{s\}, and they will not be extended at any later step\. Assume𝒜jp−1⊆𝒜jp\\mathcal\{A\}^\{p\-1\}\_\{j\}\\subseteq\\mathcal\{A\}^\{p\}\_\{j\}for every𝒦j∈𝐒\\mathcal\{K\}\_\{j\}\\in\\mathbf\{S\}\. The relational attributes of a context of𝐒\\mathbf\{S\}at stepp\+1p\+1are built on the concepts of the stepppposets of its successors, which all belong to𝐒\\mathbf\{S\}\. By the induction hypothesis applied to these successors, their posets contain all the concepts of the corresponding step\(p−1\)\(p\-1\)posets, thus every relational attribute present at stepppis still present at stepp\+1p\+1: each context of𝐒\\mathbf\{S\}only gains attributes between stepsppandp\+1p\+1\. Then𝒜jp⊆𝒜jp\+1\\mathcal\{A\}^\{p\}\_\{j\}\\subseteq\\mathcal\{A\}^\{p\+1\}\_\{j\}for every𝒦j∈𝐒\\mathcal\{K\}\_\{j\}\\in\\mathbf\{S\}\. Consequently, all successors of any context of identified objects in𝐒\\mathbf\{S\}poset\-grow monotonically, this context admits thus a concept\-poset fixpoint \(Theorem[5](https://arxiv.org/html/2609.00054#Thmtheorem5)\)\. ∎

### 5\.3Ensuring convergence in RCA\-AOC

Figure 16:Example of dependency graph\.Considering Theorems[2](https://arxiv.org/html/2609.00054#Thmtheorem2)and[5](https://arxiv.org/html/2609.00054#Thmtheorem5), we can elaborate a simple process that modifies the dataset to ensure convergence of the application of RCA\-AOC for any given RCF:

1. 1\.Detect circuits and contexts without successors \(sinks\)\.
2. 2\.Add identifiers to contexts in circuits and to every context reachable from a circuit \(i\.e\., all its transitive successors\), except sink contexts\.

The dependency graph depicted in Fig\.[16](https://arxiv.org/html/2609.00054#S5.F16)will help illustrate the process and why it leads to convergence in RCA\-AOC\. In this graph,K4K\_\{4\}andK7K\_\{7\}have no successor, thus they reach their fixpoint at the first step\. Let us now consider the contexts that are in a circuit or reachable from one\. We denote by𝐒\\mathbf\{S\}the set composed of the contexts in circuits, together with all their transitive successors \(including sinks\)\. In Fig\.[16](https://arxiv.org/html/2609.00054#S5.F16),𝐒=\{K2,K3,K4,K5,K6,K7\}\\mathbf\{S\}=\\\{K\_\{2\},K\_\{3\},K\_\{4\},K\_\{5\},K\_\{6\},K\_\{7\}\\\}\. After applying the process above, every context of𝐒\\mathbf\{S\}is either a sink or a context of identified objects and thus will admit a fixpoint by reference to Corollary[6](https://arxiv.org/html/2609.00054#Thmtheorem6)\. The remaining context \(K1K\_\{1\}in Fig\.[16](https://arxiv.org/html/2609.00054#S5.F16)\) has no predecessor and is not in a circuit nor reachable from one\. Its only successorK2K\_\{2\}belongs to𝐒\\mathbf\{S\}, hence admits a concept\-poset fixpoint\. Applying Theorem[2](https://arxiv.org/html/2609.00054#Thmtheorem2),K1K\_\{1\}has thus a fixpoint\. Note thatK1K\_\{1\}being a source context, with no predecessor, it has no effect on other contexts\.

To sum up, identifiers are added to contextsK2K\_\{2\},K3K\_\{3\},K5K\_\{5\}andK6K\_\{6\}; the other contexts admit fixpoints as they are either sinks \(K4K\_\{4\},K7K\_\{7\}\) or have all their successors admitting fixpoints \(K1K\_\{1\}\)\.

Applying this process to the UML example, and thus adding identifiers in the Operation context \(the only context where they are needed for ensuring convergence\), would ensure a converging process\.

To check if a context contains only identified objects, it is sufficient to verify that∀o∈G\\forall o\\in G,\{o\}′′=\{o\}\\\{o\\\}^\{\\prime\\prime\}=\\\{o\\\}\. In the case of unidentified objects, one can add identifiers as described previously\. In an automatic process, we propose to add identifiers without checking for two reasons: the first reason being that the cost of adding an identifying attribute is small in the AOC\-poset computation; the second reason being that when applying RCA\-AOC to different RCFs sharing the same dependency graph, the user can predict when the system will add identifiers and treat them consistently\. The only drawback from adding identifiers is that each object is introduced alone in a concept, leading to more concepts than initially wanted\. But the information added from identifiers can be easily spotted and removed from the analysis of the final AOC\-posets if needed\.

### 5\.4A converging approach: RCA\-AOC\-conv

In this section, we present a converging variant of RCA\-AOC, called RCA\-AOC\-conv\. It relies on the property that ensures the convergence of the RCA process: the concept lattice of a context𝒦in\{\\cal K\}^\{n\}\_\{i\}at stepnnis included, under the extent inclusion, in the concept lattice of𝒦in\+1\{\\cal K\}^\{n\+1\}\_\{i\}at stepn\+1n\+1\. The number of concepts of a context being bounded by the powerset of its object set, if this number is monotonically increasing then convergence is ensured\. From this same property, an implementation of this process may use an optimization that consists in considering at each step the concept lattice from the previous step and computing only new concepts generated by new attributes, as in Galicia, one of the first implementations supporting RCA\([Valtchev et al\., 2003](https://arxiv.org/html/2609.00054#bib.bib13)\)\. This is of course correct, but it cannot be adapted to RCA\-AOC as we define it because the AOC\-poset of a context at one step may not be included in the AOC\-poset of the next step\.

The process RCA\-AOC\-conv is inspired by this optimization\. At each step, the relational attributes built on the concepts of the previous concept\-posets are accumulated into the extended contexts, and the concept\-poset of a context is the AOC\-poset of its cumulated extended context \(see Algorithm[1](https://arxiv.org/html/2609.00054#alg1)in Appendix[B](https://arxiv.org/html/2609.00054#A2)\)\. As a consequence, new introducer concepts appear, while concepts that no longer introduce any element disappear\. The convergence is guaranteed, but the invariant differs from the one of RCA: here,*relational attributes*, rather than concepts, are never removed\. When a concept disappears, the relational attributes built on it at previous steps are kept, together with their incidence, which is fixed at creation \(Definition[10](https://arxiv.org/html/2609.00054#Thmdefinition10)\)\. We call such attributes*dangling attributes*: they refer to a concept that no longer belongs to the current concept\-posets\. For example, conceptco\_8of step 0 of the UML example \(Fig\.[10](https://arxiv.org/html/2609.00054#S4.F10)\) disappears at step 1, as it is no longer an object introducer; the relational attribute∃\\exists\_ownedOperation\(co\_8\), built at step 1, is nevertheless kept at all the following steps, where it refers toco\_8as a dangling attribute\.

Since relational attributes are only ever added and their incidence never changes, the successive extended contexts of a context𝒦i\\mathcal\{K\}\_\{i\}form an increasing sequence for the context inclusion defined in Sect\.[5\.2](https://arxiv.org/html/2609.00054#S5.SS2):𝒦ip⊆𝒦ip\+1\\mathcal\{K\}\_\{i\}^\{p\}\\subseteq\\mathcal\{K\}\_\{i\}^\{p\+1\}for every steppp\. Lemma[1](https://arxiv.org/html/2609.00054#Thmtheorem1)applies and RCA\-AOC\-conv converges on any RCF\.

Because non\-introducer concepts are removed, by construction, the structure built at stepppis exactly the AOC\-poset of the extended context𝒦ip\\mathcal\{K\}\_\{i\}^\{p\}\. What is sacrificed is thus not the AOC\-poset structure itself, but two properties of lattice\-based RCA\. First, the step\-to\-step inclusion of the structures is lost: the concept\-poset of one step may not be included in the concept\-poset of the next step, since a concept that becomes a non\-introducer disappears \(see the example ofC\_K1\_3in Appendix[B](https://arxiv.org/html/2609.00054#A2)\)\. Second, the result is no longer self\-contained: at the fixpoint, concept intents may contain dangling attributesρ⁡\(r\)​r​\(C\)\\rho\(r\)\\,r\(C\)whereCCbelongs to none of the final concept\-posets\. Such attributes remain interpretable, since the extent ofCCboth identifies the attribute \(Definition[10](https://arxiv.org/html/2609.00054#Thmdefinition10)\) and determines its incidence\.

An alternative convergent variant would instead keep the concepts, as in RCA: non\-introducer concepts, such asco\_8at step 1, would remain in the structure\. The step\-to\-step inclusion of the structures would then be preserved and no dangling attribute would appear, but the conceptual structure would no longer be an AOC\-poset in the general case, and the concept number would grow from the AOC\-poset size towards, at worst, the concept lattice size\. We adopt the attribute\-keeping variant, which preserves the AOC\-poset structure and its compactness while ensuring convergence: with respect to the AOC\-posets that RCA\-AOC builds on the same extended contexts, the only additional concepts are the attribute\-concepts introducing dangling attributes, which often have an empty simplified extent \(see e\.g\. conceptC\_K3\_13in Appendix[B](https://arxiv.org/html/2609.00054#A2)\)\.

In RCA\-AOC, some disappearing concepts may have little interest, as in the case of conceptco\_8of step 0 \(that groups together all operations\), but some others may be very relevant, e\.g\. conceptscpr\_22\(Fig\.[12](https://arxiv.org/html/2609.00054#S4.F12)\) andcc\_17\(Fig\.[11](https://arxiv.org/html/2609.00054#S4.F11)\) that represent respectively the new suggested propertyadminId:Stringwhose addition brings the suggestion of the new classFinancialStructure\. These concepts, successively derived from one another and alternately appearing and disappearing in RCA\-AOC, are summarized in Fig\.[17](https://arxiv.org/html/2609.00054#S5.F17)\. RCA\-AOC\-conv retains them:cc\_17is the attribute\-concept of the kept attribute∃\\exists\_ownedOperation\(co\_8\), andcpr\_22the attribute\-concept of∃\\exists\_class\(cc\_17\); both persist at every step following their creation and belong to the final result\.

Figure 17:Bank example \- Concepts created and disappearing along the steps of RCA\-AOC:co\_8present only at step 0;cc\_17present at step 1, created fromco\_8, and absent at step 2;cpr\_22present at step 2, created fromcc\_17, and absent at step 3\. In RCA\-AOC\-conv,co\_8also disappears at step 1, butcc\_17andcpr\_22, being the attribute\-concepts of the kept relational attributes built onco\_8andcc\_17respectively, persist from their creation on\.RCA\-AOC\-conv is also useful when relational attributes are developed by unfolding the concept references, to avoid having to keep the AOC\-posets of all steps\([Gutierrez et al\., 2025](https://arxiv.org/html/2609.00054#bib.bib41)\)\. RCA\-AOC and RCA\-AOC\-conv are implemented in fca4j as options of theRCAcommand\.

Only RCA\-AOC is implemented in RCAexplore\. In this software, the 2015 version does not implement any divergence check, letting the user choose between stopping manually or stopping automatically when the number of concepts stays stable between two steps\. Version 26\.1 implements a repetition check to stop the process when a step is identically repeated twice444[https://forge\.icube\.unistra\.fr/dolques/RCAExplore](https://forge.icube.unistra.fr/dolques/RCAExplore)\.

## 6Related work

##### AOC\-posets in FCA

To the best of our knowledge, the AOC\-posets have been introduced by[Godin and Mili \(1993\)](https://arxiv.org/html/2609.00054#bib.bib37)in the domain of software engineering \(object\-oriented programming\)\. In their paper, the AOC\-poset is calledpruned latticeand they consider a specific case where each formal object owns a specific formal attribute \(not owned by the others\)\. Algorithms and tools for building AOC\-posets were proposed in\([Godin et al\., 1998](https://arxiv.org/html/2609.00054#bib.bib38);[Berry et al\., 2012](https://arxiv.org/html/2609.00054#bib.bib19)\)\.

The AOC\-poset has also been used in applications of FCA to non\-monotonic reasoning and domain theory\([Hitzler, 2004](https://arxiv.org/html/2609.00054#bib.bib44)\)and to produce classifications from linguistic data\([Osswald and Petersen, 2002](https://arxiv.org/html/2609.00054#bib.bib55);[Petersen, 2004](https://arxiv.org/html/2609.00054#bib.bib57)\)\. Several software engineering works have relied on specific parts of the AOC\-poset, in particular the attribute\-concept component\. It has been used, for example, to refactor class hierarchies in code reengineering\([Huchard et al\., 2000](https://arxiv.org/html/2609.00054#bib.bib45)\), and to extract feature trees from sets of products in software product lines\([Ryssel et al\., 2011](https://arxiv.org/html/2609.00054#bib.bib61)\)\.

##### RCA foundations and connections with other frameworks

Relational Concept Analysis originated from a practical knowledge\-representation problem encountered in software engineering, databases, and ontology\-based settings: discovering concepts latent in a conceptual model\. More precisely, the objective is to make explicit concepts that provide a better factorization of descriptions, reducing duplication and thus improving the overall level of abstraction of the conceptual model\. The initial motivation was to extend early FCA\-based approaches\([Godin and Mili, 1993](https://arxiv.org/html/2609.00054#bib.bib37)\)to relational models, in particular UML models\([Dao et al\., 2004](https://arxiv.org/html/2609.00054#bib.bib23)\)\. Formalizations of RCA have been proposed in\([Huchard et al\., 2007](https://arxiv.org/html/2609.00054#bib.bib46);[Rouane\-Hacène et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib60)\); here, we build on the latter\. This framework computes in an iterative manner \(with a possible stop at each step\) several concept lattices from data represented in relational format\. The concept lattices are connected by links that abstract the relations between objects\. Several operators borrowed from Description Logics are used to build links between concepts\. Relations between these various operators and the corresponding concept lattices are described in\([Braud et al\., 2018](https://arxiv.org/html/2609.00054#bib.bib22)\)\. RCA has also been studied in relation to Description Logics\([Rouane\-Hacène et al\., 2007](https://arxiv.org/html/2609.00054#bib.bib42)\)and propositionalization\([Dolques et al\., 2014](https://arxiv.org/html/2609.00054#bib.bib30)\)\.[Euzenat \(2025\)](https://arxiv.org/html/2609.00054#bib.bib31)adopts a functional view on the RCA process, and defines the acceptable solutions \(families of concept lattices\) as the common fixpoints of two functions\. He shows that the RCA process returns the least element of the set of acceptable solutions\. Beyond RCA, a review of FCA methods applied to relational data has been done by[Leutwyler et al\. \(2024\)](https://arxiv.org/html/2609.00054#bib.bib48)\.

##### RCA variants

Beyond the core formalism, RCA has been improved with specific navigation control features on the dataset structure, on the scaling operators and on the built conceptual structures\([Dolques et al\., 2013a](https://arxiv.org/html/2609.00054#bib.bib24);[Ouzerdine et al\., 2019](https://arxiv.org/html/2609.00054#bib.bib56)\)\. These features have been operationalized on the basis of the dedicated toolRCAexplore\([Dolques et al\., 2019](https://arxiv.org/html/2609.00054#bib.bib25)\)\. More recently, LLM\-based approaches have been proposed to support the interpretation of RCA results, using knowledge\-delivery mechanisms based on rewriting strategies\. These mechanisms translate relational attributes into logic\-like formulas grounded either in concept extents or in the non\-relational attributes of concept intents\([Gutierrez et al\., 2025](https://arxiv.org/html/2609.00054#bib.bib41)\)\. In parallel, Fuzzy RCA enriches the paradigm with a graded semantics\([Boffa and Murinová, 2023](https://arxiv.org/html/2609.00054#bib.bib20)\)\. RCA being based on binary relations, higher\-arity settings are considered in Polyadic RCA\([Bazin et al\., 2024](https://arxiv.org/html/2609.00054#bib.bib16)\)\. Besides, to cope with the scalability issues of RCA,[Dolques et al\. \(2013b\)](https://arxiv.org/html/2609.00054#bib.bib28)proposed a variant based on AOC\-posets, namely RCA\-AOC\.[Dolques et al\. \(2016\)](https://arxiv.org/html/2609.00054#bib.bib29)performed more specifically a comparison between RCA\-AOC and RCA based on Iceberg lattices\([Stumme et al\., 2002](https://arxiv.org/html/2609.00054#bib.bib65)\)\. We showed that RCA\-AOC was more efficient and pertinent since it allows extracting interesting and less frequent behaviors than Iceberg lattices that limit the computed concepts to the most general ones\. Such variants should also be interesting for fuzzy or Polyadic RCA, whose results are more complex than RCA\.

##### RCA Applications

RCA has been used for the analysis and modernization of UML elements, namely in class diagrams and in use case diagrams\([Arévalo et al\., 2006](https://arxiv.org/html/2609.00054#bib.bib9);[Dolques et al\., 2012](https://arxiv.org/html/2609.00054#bib.bib27);[Guédi et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib39)\)\. In\([Moha et al\., 2008](https://arxiv.org/html/2609.00054#bib.bib51)\), RCA is used to exploit relations between methods and attributes to detect and fix design defects\. Model transformations are learned from transformation examples thanks to several kinds of relations between model elements \(e\.g\. between elements inside a model, transformation links between source elements and target elements\)\([Saada et al\., 2012](https://arxiv.org/html/2609.00054#bib.bib62)\)\. In the context of Web service composition,[Azmeh et al\. \(2011a\)](https://arxiv.org/html/2609.00054#bib.bib12)used relations between tasks in an abstract task pipeline to classify Web services according to their relevance for instantiating the pipeline tasks\. Other applications can be found in ontology engineering\([Bendaoud et al\., 2008](https://arxiv.org/html/2609.00054#bib.bib18);[Rouane\-Hacène et al\., 2011](https://arxiv.org/html/2609.00054#bib.bib43)\)\. More recently, RCA was used to extract knowledge graphs from relational data about neurological examinations\([Wajnberg et al\., 2018](https://arxiv.org/html/2609.00054#bib.bib67)\), to analyze data on pediatric cancers\([Wajnberg et al\., 2020](https://arxiv.org/html/2609.00054#bib.bib69)\)or faults in aluminum die casting process\([Wajnberg et al\., 2019](https://arxiv.org/html/2609.00054#bib.bib68)\), to query legal documents\([Mimouni et al\., 2013](https://arxiv.org/html/2609.00054#bib.bib49)\), while[Nica et al\. \(2020b\)](https://arxiv.org/html/2609.00054#bib.bib53)used it to extract closed partially\-ordered patterns \(acyclic graphs\) from temporal sequences on water quality\. RCA has also been used for extracting interdependent linked keys from RDF datasets, including cyclic dependencies\([Atencia et al\., 2019](https://arxiv.org/html/2609.00054#bib.bib10);[Atencia et al\., 2020](https://arxiv.org/html/2609.00054#bib.bib11)\)\.[Semeraro et al\. \(2023\)](https://arxiv.org/html/2609.00054#bib.bib64)used FCA and RCA to extract rules from data on physical systems, for the design of digital twins in the engineering domain\. The same approach is applied in the specific case of digital twins for compressed air energy storage systems\([Semeraro et al\., 2025](https://arxiv.org/html/2609.00054#bib.bib63)\)\. In most of these applications, the existential scaling operator is used, and the datasets are medium\-sized guaranteeing the feasibility of the approach\. When dealing with larger or more complex datasets, RCA\-AOC or RCA using Iceberg lattices can be applied, e\.g\. to analyze time series from river monitoring\([Braud et al\., 2022](https://arxiv.org/html/2609.00054#bib.bib21)\)or complex data about ancient remedies\([Fokou et al\., 2024](https://arxiv.org/html/2609.00054#bib.bib35)\)\.

These works show that RCA\-AOC is relevant not only as a more compact alternative to lattice\-based RCA, but also as a way to focus on concepts that are meaningful for specific applications, including concepts that may be discarded by support\-based restrictions such as Iceberg lattices\.

## 7Conclusion

This paper addressed the convergence of Relational Concept Analysis when concept lattices are replaced by AOC\-posets, yielding the RCA\-AOC variant\. AOC\-posets provide a compact and efficient representation that helps mitigate the computational complexity associated with large datasets\. In some applications, only introducer concepts are useful, while in others they are sufficient to capture the main outcomes of the analysis\. However, replacing concept lattices with AOC\-posets generally means losing the convergence guarantee of lattice\-based RCA\. We illustrated possible divergence through three examples involving existential and strict universal scaling, including one grounded in a UML class\-model refactoring task\. We then identified properties that prevent divergence\. We discussed how convergence can be recovered through conditions on the data and on the process\. Finally, we proposed a convergent variant \(RCA\-AOC\-conv\) whose structures are the AOC\-posets of cumulated extended contexts: convergence is obtained by never removing relational attributes, some of which may end up referring to concepts absent from the final structures\.

As future work, we plan to develop practical tools to detect potential non\-convergence in real application domains and to automatically repair problematic datasets using the procedure proposed in this paper\. A key objective will be to predict divergence without having to compute entire families of concept\-posets until a repetition of step sequences is observed\. We will also examine whether it is preferable to enforce convergence by modifying the dataset \(as discussed in the previous section\) or by adopting a convergent process variant \(e\.g\. RCA\-AOC\-conv\)\. In particular, we will study the trade\-offs between these two approaches in terms of computational complexity and the size \(and usefulness\) of the resulting conceptual structures\. We plan to study the alternative convergent variant discussed in Sect\.[5\.4](https://arxiv.org/html/2609.00054#S5.SS4), which would keep the concepts that no longer introduce any element: in particular, adding attributes to these concepts so that they remain introducers, as suggested by[Aranda\-Corral et al\. \(2026\)](https://arxiv.org/html/2609.00054#bib.bib6);[Aranda\-Corral et al\. \(2024\)](https://arxiv.org/html/2609.00054#bib.bib7), would preserve both the AOC\-poset structure and the step\-to\-step inclusion of the structures, avoiding dangling attributes\. Moreover, since we can now ensure convergence for a given dataset, we expect to implement RCA\-AOC more efficiently, for instance by relying on an incremental AOC\-poset construction algorithm\. We also intend to further study how divergence relates to the choice and combination of scaling operators, in particular by characterizing which ones are more prone to trigger non\-convergence and under which data conditions\. Finally, we could extend to RCA\-AOC the functional view of[Euzenat \(2025\)](https://arxiv.org/html/2609.00054#bib.bib31), which defines the acceptable solutions \(families of concept lattices\) as the common fixpoints of two functions, where RCA returns the least element of the set of acceptable solutions\.

## Declaration of generative AI and AI\-assisted technologies in the writing process

During the preparation of this work, the authors used Claude \(Anthropic\) in order to improve the language and readability of the manuscript, to obtain feedback on the mathematical definitions, proofs, and examples, and to assist with the drafting of LaTeX code\. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article\.

## Acknowledgement

This work was supported by the French National Research Agency Grant ANR\-21\-CE23\-0023 \(SmartFCA\)\.

## References

- Aranda\-Corralet al\.\(2024\)G\.A\. Aranda\-Corral, A\. Bundy, J\. Borrego\-Díaz, and P\.Y\. ChanGrounding problem in Formal Concept Analysis by means of Large Language Models\.Note:Workshop on Late Breaking Advances on Conceptual Structures @ Concepts 2024Cited by:[§7](https://arxiv.org/html/2609.00054#S7.p2.1)\.
- Aranda\-Corralet al\.\(2026\)G\.A\. Aranda\-Corral, A\. Bundy, J\. Borrego\-Díaz, and P\.Y\. ChanGrounding problem in FCA by means of LLMs\.Note:Zenodo, Slides presented at conference Concepts 24External Links:[Link](https://zenodo.org/records/18608706)Cited by:[§7](https://arxiv.org/html/2609.00054#S7.p2.1)\.
- Arévaloet al\.\(2006\)G\. Arévalo, J\. Falleri, M\. Huchard, and C\. NebutBuilding Abstractions in Class Models: Formal Concept Analysis in a Model\-Driven Approach\.InMoDELS 2006,pp\. 513–527\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Atenciaet al\.\(2019\)M\. Atencia, J\. David, J\. Euzenat, A\. Napoli, and J\. VizziniA guided walk into link key candidate extraction with relational concept analysis\.InISWC 2019,pp\. 1–9\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Atenciaet al\.\(2020\)M\. Atencia, J\. David, J\. Euzenat, A\. Napoli, and J\. VizziniLink key candidate extraction with relational concept analysis\.Discret\. Appl\. Math\.273,pp\. 2–20\.External Links:[Document](https://dx.doi.org/10.1016/J.DAM.2019.02.012)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Atzmuelleret al\.\(2024\)M\. Atzmueller, J\. Fürnkranz, T\. Kliegr, and U\. SchmidExplainable and interpretable machine learning and data mining\.Data Min\. Knowl\. Discov\.38\(5\),pp\. 2571–2595\.External Links:ISSN 1384\-5810,[Document](https://dx.doi.org/10.1007/s10618-024-01041-y)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Azmehet al\.\(2011a\)Z\. Azmeh, M\. Driss, F\. Hamoui, M\. Huchard, N\. Moha, and C\. TibermacineSelection of Composable Web Services Driven by User Requirements\.InICWS 2011,pp\. 395–402\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Azmehet al\.\(2011b\)Z\. Azmeh, M\. Huchard, A\. Napoli, M\. Rouane\-Hacène, and P\. ValtchevQuerying relational concept lattices\.InCLA 2011: Concept Lattices and their Applications,pp\. 377–392\.External Links:ISBN 978\-2\-905267\-78\-8Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p2.1)\.
- F\. Baader, D\. Calvanese, D\. McGuinness, D\. Nardi, and P\. Patel\-Schneider \(Eds\.\) \(2003\)F\. Baader, D\. Calvanese, D\. McGuinness, D\. Nardi, and P\. Patel\-Schneider \(Eds\.\)The description logic handbook theory, implementation and applications\.Cambridge University Press,Cambridge, MA\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p2.1)\.
- Bazinet al\.\(2024\)A\. Bazin, J\. Galasso, and G\. KahnPolyadic relational concept analysis\.Int\. J\. Approx\. Reason\.164,pp\. 109067\.External Links:[Document](https://dx.doi.org/10.1016/J.IJAR.2023.109067)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Belohlávek and Vychodil \(2005\)R\. Belohlávek and V\. VychodilWhat is a fuzzy concept lattice?\.InCLA 2005: Concept Lattices and their Applications,CEUR\-WS Proc\., Vol\.162\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Bendaoudet al\.\(2008\)R\. Bendaoud, A\. Napoli, and Y\. ToussaintFormal Concept Analysis: A unified framework for building and refining ontologies\.InEKAW 2008,LNCS 5268,pp\. 156–171\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Berryet al\.\(2012\)A\. Berry, M\. Huchard, A\. Napoli, and A\. SigayretHermes: an efficient algorithm for building Galois Sub\-hierarchies\.InCLA 2012: Concept Lattices and their Applications,pp\. 21–32\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p1.1)\.
- Boffa and Murinová \(2023\)S\. Boffa and P\. MurinováLogical Relations Between T\-Scaling Quantifiers and Their Implications in Fuzzy Relational Concept Analysis\.InFuzzy Logic and Technology, and Aggregation Operators \- EUSFLAT 2023, and AGOP 2023, Proceedings,LNCS, Vol\.14069,pp\. 393–404\.External Links:[Document](https://dx.doi.org/10.1007/978-3-031-39965-7%5F33)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Braudet al\.\(2022\)A\. Braud, X\. Dolques, A\. Gutierrez, M\. Huchard, P\. Keip, F\. Le Ber, P\. Martin, C\. Nica, and P\. SilvieDealing with large volumes of complex relational data using RCA\.InComplex Data Analytics with Formal Concept Analysis,pp\. 105–134\.External Links:[Document](https://dx.doi.org/10.1007/978-3-030-93278-7%5F5)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p3.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Braudet al\.\(2018\)A\. Braud, X\. Dolques, M\. Huchard, and F\. Le BerGeneralization effect of quantifiers in a classification based on relational concept analysis\.Knowl\. Based Syst\.160,pp\. 119–135\.External Links:[Document](https://dx.doi.org/10.1016/J.KNOSYS.2018.06.011)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Buzmakovet al\.\(2014\)A\. Buzmakov, S\. O\. Kuznetsov, and A\. NapoliIs concept stability a measure for pattern selection?\.Procedia Computer Science31,pp\. 918–927\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p2.1)\.
- Daoet al\.\(2004\)M\. Dao, M\. Huchard, M\. Rouane\-Hacène, C\. Roume, and P\. ValtchevImproving generalization level in UML models iterative cross generalization in practice\.InConceptual Structures at Work: ICCS 2004,LNCS, Vol\.3127,pp\. 346–360\.External Links:[Document](https://dx.doi.org/10.1007/978-3-540-27769-9%5F23)Cited by:[§4\.1](https://arxiv.org/html/2609.00054#S4.SS1.p1.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Dolqueset al\.\(2019\)X\. Dolques, A\. Braud, M\. Huchard, and F\. Le BerRCAexplore, a FCA based tool to explore relational data\.InSupplementary Proceedings of ICFCA 2019 Conference,CEUR\-WS Proc\., Vol\.2378,pp\. 55–59\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Dolqueset al\.\(2010\)X\. Dolques, M\. Huchard, C\. Nebut, and P\. ReitzLearning transformation rules from transformation examples: an approach based on relational concept analysis\.In2010 14th IEEE International Enterprise Distributed Object Computing Conference Workshops,pp\. 27–32\.Cited by:[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p1.1)\.
- Dolqueset al\.\(2012\)X\. Dolques, M\. Huchard, C\. Nebut, and P\. ReitzFixing Generalization Defects in UML Use Case Diagrams\.Fundam\. Inform\.115\(4\),pp\. 327–356\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Dolqueset al\.\(2016\)X\. Dolques, F\. Le Ber, M\. Huchard, and C\. GracPerformance\-friendly rule extraction in large water data\-sets with AOC posets and relational concept analysis\.International Journal of General Systems45\(1\),pp\. 1–24\.External Links:http://dx\.doi\.org/10\.1080/03081079\.2015\.1072927Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p3.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Dolqueset al\.\(2013a\)X\. Dolques, F\. Le Ber, M\. Huchard, and C\. NebutRelational concept analysis for relational data exploration, vol\. 5\.InAdvances in Knowledge Discovery and Management,SCI, Vol\.615,pp\. 57–77\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Dolqueset al\.\(2013b\)X\. Dolques, F\. Le Ber, and M\. HuchardAOC\-Posets: a Scalable Alternative to Concept Lattices for Relational Concept Analysis\.InCLA 2013: Concept Lattices and their Applications,CEUR\-WS Proc\.,pp\. 129–140\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p2.1),[§3](https://arxiv.org/html/2609.00054#S3.p1.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Dolqueset al\.\(2014\)X\. Dolques, K\. C\. Mondal, A\. Braud, M\. Huchard, and F\. Le BerRCA as a data transforming method: A comparison with propositionalisation\.InFormal Concept Analysis \- ICFCA 2014,LNCS, Vol\.8478,pp\. 112–127\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Euzenat \(2025\)J\. EuzenatThe fixed\-point semantics of relational concept analysis\.J\. Artif\. Intell\. Res\.83\.External Links:[Document](https://dx.doi.org/10.1613/JAIR.1.17882)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1),[§7](https://arxiv.org/html/2609.00054#S7.p2.1)\.
- Ferré and Cellier \(2018\)S\. Ferré and P\. CellierHow Hierarchies of Concept Graphs Can Facilitate the Interpretation of RCA Lattices?\.InCLA 2018 : Concept Lattices and their Applications,CEUR\-WS Proc\., Vol\.2123\.Cited by:[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p1.1)\.
- Ferré and Cellier \(2020\)S\. Ferré and P\. CellierGraph\-FCA: An extension of formal concept analysis to knowledge graphs\.Discrete applied mathematics273,pp\. 81–102\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1),[§1](https://arxiv.org/html/2609.00054#S1.p2.1),[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p2.1)\.
- Fokouet al\.\(2025\)V\. Fokou, P\. Cellier, X\. Dolques, S\. Ferré, and F\. Le BerTheoretical comparison of Relational Concept Analysis \(RCA\) and Graph\-FCA \(GCA\)\.Int\. Journal of Approximate Reasoning186,pp\. 109496\.External Links:[Document](https://dx.doi.org/10.1016/j.ijar.2025.109496)Cited by:[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p2.1),[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p3.1)\.
- Fokouet al\.\(2024\)V\. Fokou, K\. El Haff, A\. Braud, X\. Dolques, F\. Le Ber, and V\. PitchonExploring Old Arabic Remedies with Formal and Relational Concept Analysis\.InConceptual Knowledge Structure \- Concepts 2024, Proceedings,Cited by:[§2\.2](https://arxiv.org/html/2609.00054#S2.SS2.p4.1),[§2](https://arxiv.org/html/2609.00054#S2.p1.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Ganter and Wille \(1999\)B\. Ganter and R\. WilleFormal concept analysis: mathematical foundations\.Springer Verlag\.Cited by:[Appendix A](https://arxiv.org/html/2609.00054#A1.p4.1.1),[§1](https://arxiv.org/html/2609.00054#S1.p1.1),[§2\.1](https://arxiv.org/html/2609.00054#S2.SS1.p1.1)\.
- Godinet al\.\(1998\)R\. Godin, H\. Mili, G\. W\. Mineau, R\. Missaoui, A\. Arfi, and T\.\-T\. ChauDesign of Class Hierarchies based on Concept \(Galois\) Lattices\.Theory and Practice of Object Systems4\(2\),pp\. 117–134\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p1.1)\.
- Godin and Mili \(1993\)R\. Godin and H\. MiliBuilding and Maintaining Analysis\-Level Class Hierarchies using Galois Lattices\.InOOPSLA ’93,Vol\.28,pp\. 394–410\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p2.1),[§4\.1](https://arxiv.org/html/2609.00054#S4.SS1.p1.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p1.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Guédiet al\.\(2013\)A\. O\. Guédi, A\. Miralles, M\. Huchard, and C\. NebutA practical application of relational concept analysis to class model factorization: lessons learned from a thematic information system\.InCLA 2013 : Concept Lattices and Their Applications,CEUR\-WS Proc\., Vol\.1062,pp\. 9–20\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Guenouneet al\.\(2025\)H\. Guenoune, A\. Gutierrez, M\. Huchard, M\. Lafourcade, P\. Martin, A\. Miralles, and H\. ZhangLLM\-Assisted Relational Concept Analysis for Class Model Restructuring\.InConceptual Knowledge Structures \- CONCEPTS 2025, Proceedings,LNCS, Vol\.15941,pp\. 107–123\.External Links:[Document](https://dx.doi.org/10.1007/978-3-032-03364-2%5F7)Cited by:[§4\.1](https://arxiv.org/html/2609.00054#S4.SS1.p1.1),[§4\.1](https://arxiv.org/html/2609.00054#S4.SS1.p17.1)\.
- Guidottiet al\.\(2018\)R\. Guidotti, A\. Monreale, S\. Ruggieri, F\. Turini, F\. Giannotti, and D\. PedreschiA survey of methods for explaining black box models\.ACM Comput\. Surv\.51\(5\)\.External Links:ISSN 0360\-0300,[Document](https://dx.doi.org/10.1145/3236009)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Gutierrezet al\.\(2025\)A\. Gutierrez, M\. Huchard, P\. Martin, and H\. ZhangEmpowering relational concept analysis using large language model knowledge delivery\.InConceptual Knowledge Structures \- CONCEPTS 2025, Proceedings,LNCS, Vol\.15941,pp\. 124–139\.Cited by:[§4](https://arxiv.org/html/2609.00054#S4.p2.1),[§5\.4](https://arxiv.org/html/2609.00054#S5.SS4.p7.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Hastieet al\.\(2009\)T\. Hastie, R\. Tibshirani, and J\. FriedmanThe elements of statistical learning: data mining, inference, and prediction\.2 edition,Springer\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Hitzler \(2004\)P\. HitzlerDefault Reasoning over Domains and Concept Hierarchies\.InKI 2004: Advances in Artificial Intelligence,LNCS, Vol\.3238,pp\. 351–365\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p2.1)\.
- Huchardet al\.\(2000\)M\. Huchard, H\. Dicky, and H\. LeblancGalois Lattice as a Framework to specify Algorithms Building Class Hierarchies\.Theoretical Informatics and Applications34,pp\. 521–548\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p2.1)\.
- Huchardet al\.\(2007\)M\. Huchard, M\. Rouane\-Hacène, C\. Roume, and P\. ValtchevRelational concept discovery in structured datasets\.Ann\. Math\. Artif\. Intell\.49\(1\-4\),pp\. 39–76\.Cited by:[§2\.2](https://arxiv.org/html/2609.00054#S2.SS2.p1.1),[§4\.1](https://arxiv.org/html/2609.00054#S4.SS1.p1.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Kötters and Eklund \(2020\)J\. Kötters and P\. W\. EklundConjunctive query pattern structures: A relational database model for formal concept analysis\.Discret\. Appl\. Math\.273,pp\. 144–171\.External Links:[Document](https://dx.doi.org/10.1016/J.DAM.2019.08.019)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1),[§1](https://arxiv.org/html/2609.00054#S1.p2.1)\.
- Kuznetsov and Makhalova \(2018\)S\. O\. Kuznetsov and T\. P\. MakhalovaOn interestingness measures of formal concepts\.Inf\. Sci\.442\-443,pp\. 202–219\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p2.1)\.
- LeCunet al\.\(2015\)Y\. LeCun, Y\. Bengio, and G\. HintonDeep learning\.Nature521\(7553\),pp\. 436–444\.External Links:[Document](https://dx.doi.org/10.1038/nature14539),ISBN 1476\-4687Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Leutwyleret al\.\(2024\)N\. Leutwyler, M\. Lezoche, C\. Franciosi, H\. Panetto, L\. Teste, and D\. TorresMethods for concept analysis and multi\-relational data mining: a systematic literature review\.Knowl\. Inf\. Syst\.66\(9\),pp\. 5113–5150\.External Links:[Document](https://dx.doi.org/10.1007/S10115-024-02139-X)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Mimouniet al\.\(2013\)N\. Mimouni, M\. Fernández, A\. Nazarenko, D\. Bourcier, and S\. SalottiA relational approach for information retrieval on XML legal sources\.InInternational Conference on Artificial Intelligence and Law, ICAIL’13,E\. Francesconi and B\. Verheij \(Eds\.\),pp\. 212–216\.External Links:[Document](https://dx.doi.org/10.1145/2514601.2514629)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Miralleset al\.\(2015\)A\. Miralles, G\. Molla, M\. Huchard, C\. Nebut, L\. Deruelle, and M\. DerrasClass model normalization \- outperforming formal concept analysis approaches with aoc\-posets\.InCLA 2015 : Concept Lattices and Their Applications,CEUR\-WS Proc\., Vol\.1466,pp\. 111–122\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p3.1),[§4\.1](https://arxiv.org/html/2609.00054#S4.SS1.p1.1)\.
- Mohaet al\.\(2008\)N\. Moha, M\. Rouane\-Hacène, P\. Valtchev, and Y\. GuéhéneucRefactorings of Design Defects Using Relational Concept Analysis\.InFormal Concept Analysis, ICFCA 2008,LNAI, Vol\.4933,pp\. 289–304\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Nicaet al\.\(2020a\)C\. Nica, V\. Almăşan, and A\. GrozaFastRCA\-Seq: an efficient approach for extracting hierarchies of multilevel closed partially\-ordered patterns\.Knowledge\-Based Systems210,pp\. 106533\.External Links:ISSN 0950\-7051,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.knosys.2020.106533)Cited by:[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p1.1),[§4](https://arxiv.org/html/2609.00054#S4.p2.1)\.
- Nicaet al\.\(2020b\)C\. Nica, A\. Braud, and F\. Le BerRCA\-Seq: an Original Approach for Enhancing the Analysis of Sequential Data Based on Hierarchies of Multilevel Closed Partially\-Ordered Patterns\.Discrete Applied Mathematics273,pp\. 232–251\.External Links:[Document](https://dx.doi.org/10.1016/j.dam.2019.02.037)Cited by:[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p1.1),[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p2.1),[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p3.1),[§4](https://arxiv.org/html/2609.00054#S4.p2.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Object Management Group \(2017\)Object Management GroupUnified Modeling Language, Version 2\.5\.1\.Note:OMG Document Number formal/2017\-12 \([https://www\.omg\.org/spec/UML/2\.5\.1](https://www.omg.org/spec/UML/2.5.1)\)Cited by:[§4\.1](https://arxiv.org/html/2609.00054#S4.SS1.p3.1)\.
- Osswald and Petersen \(2002\)R\. Osswald and W\. PetersenInduction of Classifications from Linguistic Data\.InECAI’02 Workshop,Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p2.1)\.
- Ouzerdineet al\.\(2019\)A\. Ouzerdine, A\. Braud, X\. Dolques, M\. Huchard, and F\. Le BerAdjusting the exploration flow in relational concept analysis \- an experience on a watercourse quality dataset\.InAdvances in Knowledge Discovery and Management, Vol\. 9,SCI, Vol\.1004,pp\. 175–198\.External Links:[Document](https://dx.doi.org/10.1007/978-3-030-90287-2%5F9)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Petersen \(2004\)W\. PetersenA set\-theoretical approach for the induction of inheritance hierarchies\.Electronic Notes in Theoretical Computer Science53,pp\. 296–308\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p2.1)\.
- Poelmanset al\.\(2013a\)J\. Poelmans, D\. I\. Ignatov, S\. O\. Kuznetsov, and G\. DedeneFormal concept analysis in knowledge processing: A survey on applications\.Expert Syst\. Appl\.40\(16\),pp\. 6538–6560\.External Links:[Document](https://dx.doi.org/10.1016/J.ESWA.2013.05.009)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Poelmanset al\.\(2013b\)J\. Poelmans, S\. O\. Kuznetsov, D\. I\. Ignatov, and G\. DedeneFormal concept analysis in knowledge processing: A survey on models and techniques\.Expert Syst\. Appl\.40\(16\),pp\. 6601–6623\.External Links:[Document](https://dx.doi.org/10.1016/J.ESWA.2013.05.007)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Rouane\-Hacèneet al\.\(2007\)M\. Rouane\-Hacène, M\. Huchard, A\. Napoli, and P\. ValtchevA proposal for combining formal concept analysis and description logics for mining relational data\.InFormal Concept Analysis, ICFCA 2007, Proceedings,LNCS, Vol\.4390,pp\. 51–65\.External Links:[Document](https://dx.doi.org/10.1007/978-3-540-70901-5%5F4)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Rouane\-Hacèneet al\.\(2013\)M\. Rouane\-Hacène, M\. Huchard, A\. Napoli, and P\. ValtchevRelational concept analysis: mining concept lattices from multi\-relational data\.Ann\. Math\. Artif\. Intell\.67\(1\),pp\. 81–108\.Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1),[§1](https://arxiv.org/html/2609.00054#S1.p2.1),[§2\.2](https://arxiv.org/html/2609.00054#S2.SS2.p1.1),[§2\.3](https://arxiv.org/html/2609.00054#S2.SS3.p1.1),[§3](https://arxiv.org/html/2609.00054#S3.p2.1),[§4](https://arxiv.org/html/2609.00054#S4.p2.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px2.p1.1)\.
- Rouane\-Hacèneet al\.\(2011\)M\. Rouane\-Hacène, P\. Valtchev, and R\. NkambouSupporting Ontology Design through Large\-Scale FCA\-Based Ontology Restructuring\.InICCS 2011,pp\. 257–269\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Rysselet al\.\(2011\)U\. Ryssel, J\. Ploennigs, and K\. KabitzschExtraction of feature models from formal contexts\.InSPLC 2011 Workshops,Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px1.p2.1)\.
- Saadaet al\.\(2012\)H\. Saada, X\. Dolques, M\. Huchard, C\. Nebut, and H\. A\. SahraouiGeneration of Operational Transformation Rules from Examples of Model Transformations\.InMoDELS 2012, Model Driven Engineering Languages and Systems,LNCS, Vol\.7590,pp\. 546–561\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Semeraroet al\.\(2025\)C\. Semeraro, R\. F\. Ababneh, L\. A\. Alkhatib, D\. Saqallah, R\. Al Koutoubi, H\. Aljaghoub, A\. H\. Alami, M\. A\. Abdelkareem, and A\. G\. OlabiData\-driven digital twin for fault detection in compressed air energy storage systems: design and experimental validation\.Energy,pp\. 138401\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Semeraroet al\.\(2023\)C\. Semeraro, M\. Lezoche, H\. Panetto, and M\. DassistiData\-driven invariant modelling patterns for digital twin design\.Journal of Industrial Information Integration31,pp\. 100424\.External Links:ISSN 2452\-414X,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.jii.2022.100424)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Stummeet al\.\(2002\)G\. Stumme, R\. Taouil, Y\. Bastide, N\. Pasquier, and L\. LakhalComputing iceberg concept lattices with titanic\.Data Knowl\. Eng\.42\(2\),pp\. 189–222\.External Links:ISSN 0169\-023X,[Document](https://dx.doi.org/10.1016/S0169-023X%2802%2900057-5)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p2.1),[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px3.p1.1)\.
- Valtchevet al\.\(2003\)P\. Valtchev, D\. Grosser, C\. Roume, and M\. Rouane\-HacèneGalicia : an open platform for lattices\.External Links:[Link](https://api.semanticscholar.org/CorpusID:16650318)Cited by:[§5\.4](https://arxiv.org/html/2609.00054#S5.SS4.p1.1)\.
- Voutsadakis \(2002\)G\. VoutsadakisPolyadic concept analysis\.Order19\(3\),pp\. 295–304\.External Links:[Document](https://dx.doi.org/10.1023/A%3A1021252203599)Cited by:[§1](https://arxiv.org/html/2609.00054#S1.p1.1)\.
- Wajnberget al\.\(2018\)M\. Wajnberg, M\. Lezoche, A\. Blondin\-Massé, P\. Valtchev, H\. Panetto, and L\. TyvaertSemantic interoperability of large systems through a formal method: relational concept analysis\.IFAC\-PapersOnLine51\(11\),pp\. 1397–1402\.Note:Special Issue: 16th IFAC Symposium on Information Control Problems in Manufacturing INCOM 2018External Links:ISSN 2405\-8963,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.ifacol.2018.08.330)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Wajnberget al\.\(2019\)M\. Wajnberg, P\. Valtchev, M\. Lezoche, A\. B\. Massé, and H\. PanettoConcept analysis\-based association mining from linked data: A case in industrial decision making\.InProceedings of the Joint Ontology Workshops 2019 Episode V: The Styrian Autumn of Ontology,CEUR\-WS Proc\., Vol\.2518\.Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.
- Wajnberget al\.\(2020\)M\. Wajnberg, P\. Valtchev, A\. B\. Massé, A\. Benmoussa, M\. Krajinovic, C\. Laverdière, E\. Levy, D\. Sinnett, and V\. MarcilMining heterogeneous associations from pediatric cancer data by relational concept analysis\.InICDM Workshops 2020,pp\. 597–604\.External Links:[Document](https://dx.doi.org/10.1109/ICDMW51313.2020.00085)Cited by:[§6](https://arxiv.org/html/2609.00054#S6.SS0.SSS0.Px4.p1.1)\.

## Appendix AProof of Lemma[4](https://arxiv.org/html/2609.00054#Thmtheorem4)

###### Proof\.

Figure 18:Illustration of the incremental construction of an AOC\-poset by adding an attribute\. Each concept is represented with its full intent and extent\. The simplified intent and simplified extent elements are circled\. Concepts represented with dashed lines are concepts from the lattice that do not belong to the AOC\-poset in the pictured cases: the conceptsC∗C^\{\*\}from cases 3\.c and 4\.c can never exist in the AOC\-poset while the conceptsC2C\_\{2\}from cases 4\.a and 4\.b could belong to the AOC\-poset if𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)∩𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C​A\)≠∅\\mathit\{Extent\}\_\{S\}\(C\)\\cap\\mathit\{Extent\}\(CA\)\\neq\\emptyset\. Crossed\-out concepts are concepts that cannot exist even in the concept lattice\.We begin with a few notations\. Let𝒦=\(G,M,I\)\\mathcal\{K\}=\(G,M,I\)be a context and let𝒜\\mathcal\{A\}be its AOC\-poset\. Let𝑎𝑡𝑡\\mathit\{att\}denote a new attribute andg⊆G\\mathit\{g\}\\subseteq Gthe set of objects owning this attribute\.𝒦∗=\(G,M∪\{𝑎𝑡𝑡\},I∪g×\{𝑎𝑡𝑡\}\)\\mathcal\{K\}^\{\*\}=\(G,M\\cup\\\{\\mathit\{att\}\\\},I\\cup g\\times\\\{\\mathit\{att\}\\\}\)is the context resulting from the addition of the attribute𝑎𝑡𝑡\\mathit\{att\}to the objects ofg\\mathit\{g\}in𝒦\\mathcal\{K\}and𝒜∗\\mathcal\{A\}^\{\*\}is its AOC\-poset\.𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)\\mathit\{Intent\}\_\{S\}\(C\)denotes the simplified intent ofCC,𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)\\mathit\{Extent\}\_\{S\}\(C\)its simplified extent\.

Then we introduce two lemmas\. The first one states that the property of being a context of identified objects is preserved by a single attribute addition\. The second one states that𝒜∗\\mathcal\{A\}^\{\*\}necessarily contains a concept whose extent isgg, namely the attribute\-concept of𝑎𝑡𝑡\\mathit\{att\}, denoted𝐶𝐴\\mathit\{CA\}in the following\.

###### Lemma 7\(Preservation\)\.

If𝒦\\mathcal\{K\}is a context of identified objects, then𝒦∗\\mathcal\{K\}^\{\*\}is also a context of identified objects\.

###### Proof\.

Leto∈Go\\in G, and let\(⋅\)′⁣∗\(\\cdot\)^\{\\prime\*\}denote derivation in𝒦∗\\mathcal\{K\}^\{\*\}\. Since\{o\}′⁣∗∩M=\{o\}′\\\{o\\\}^\{\\prime\*\}\\cap M=\\\{o\\\}^\{\\prime\}, every object owning all attributes of\{o\}′⁣∗\\\{o\\\}^\{\\prime\*\}owns in particular all attributes of\{o\}′\\\{o\\\}^\{\\prime\}, hence\{o\}⊆\{o\}′′∗⊆\{o\}′′=\{o\}\\\{o\\\}\\subseteq\\\{o\\\}^\{\\prime\\prime\*\}\\subseteq\\\{o\\\}^\{\\prime\\prime\}=\\\{o\\\}, and therefore\{o\}′′∗=\{o\}\\\{o\\\}^\{\\prime\\prime\*\}=\\\{o\\\}\. ∎

###### Lemma 8\(Existence of an attribute\-concept CA\)\.

The setggis closed in𝒦∗\\mathcal\{K\}^\{\*\}, and𝐶𝐴=\(g,g′\)\\mathit\{CA\}=\(g,\\,g^\{\\prime\}\)is a formal concept of𝒦∗\\mathcal\{K\}^\{\*\}, namely the attribute\-concept of𝑎𝑡𝑡\\mathit\{att\}\. In particular,𝑎𝑡𝑡∈𝐼𝑛𝑡𝑒𝑛𝑡S​\(𝐶𝐴\)\\mathit\{att\}\\in\\mathit\{Intent\}\_\{S\}\(\\mathit\{CA\}\)and𝐶𝐴\\mathit\{CA\}belongs to𝒜∗\\mathcal\{A\}^\{\*\}\.

###### Proof\.

The setggis closed in𝒦∗\\mathcal\{K\}^\{\*\}since\{a​t​t\}′=g\\\{att\\\}^\{\\prime\}=gby construction \(any image by a derivation operator is closed\([Ganter and Wille, 1999](https://arxiv.org/html/2609.00054#bib.bib36)\)\)\.𝐶𝐴=\(g,g′\)\\mathit\{CA\}=\(g,g^\{\\prime\}\)is thus a well\-defined concept of𝒦∗\\mathcal\{K\}^\{\*\}whose simplified intent containsa​t​tatt\. ∎

For eachC=\(X,Y\)∈𝒜C=\(X,Y\)\\in\\mathcal\{A\}, we now exhaustively study the different cases that may occur\. Those cases are illustrated by Fig\.[18](https://arxiv.org/html/2609.00054#A1.F18)to help the reader\. During our reasoning we consider that the identifier of a concept is its extent and that two concepts from different AOC\-posets are equivalent if their extents are the same regardless of their intent\. We denote the equivalence relation between those concepts by the binary operator∼\\sim\. Relational attributes are identified as stated in Definition[10](https://arxiv.org/html/2609.00054#Thmdefinition10): if a concept equivalent toCCexists at the following step, the attributeρ⁡\(r\)​r​\(C\)\\rho\(r\)\\,r\(C\)is considered to be the same, and its incidence is unchanged, since it only depends on\(ρ⁡\(r\),r,𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)\)\(\\rho\(r\),r,\\mathit\{Extent\}\(C\)\)\.

1. 1\.ifg=Xg=XthenC∼C​AC\\sim CA\. In this situation, the considered concept is equivalent to the concept introducingatt, soCAis built just by addingattto the intent ofC\.
2. 2\.ifX⊂gX\\subset g,∃C∗=\(X,Y∪\{a​t​t\}\)∈𝒜∗\\exists C^\{\*\}=\(X,Y\\cup\\\{att\\\}\)\\in\\mathcal\{A\}^\{\*\}andC∼C∗C\\sim C^\{\*\}\. In other words, for every concept whose extent is strictly included ing,attis added to its intent\. It becomes more specific thanCA, but it keeps its simplified intent and simplified extent\.
3. 3\.ifg⊂Xg\\subset X, 1. \(a\)If𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)≠∅\\mathit\{Intent\}\_\{S\}\(C\)\\neq\\emptysetthen∃C∗=\(X,Y\)∈𝒜∗\\exists C^\{\*\}=\(X,Y\)\\in\\mathcal\{A\}^\{\*\}as𝐼𝑛𝑡𝑒𝑛𝑡S​\(C∗\)=𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)\\mathit\{Intent\}\_\{S\}\(C^\{\*\}\)=\\mathit\{Intent\}\_\{S\}\(C\)\. This corresponds to the concepts whose extent containsgand which have a non\-empty simplified intent\. The conceptCCremains asC∗C^\{\*\}with the same simplified intent, the simplified extent may be reduced, and become𝐸𝑥𝑡𝑒𝑛𝑡S​\(C∗\)=𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)∖g\\mathit\{Extent\}\_\{S\}\(C^\{\*\}\)=\\mathit\{Extent\}\_\{S\}\(C\)\\setminus g, if𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)∩g≠∅\\mathit\{Extent\}\_\{S\}\(C\)\\cap g\\neq\\emptyset\. 2. \(b\)If𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)=∅\\mathit\{Intent\}\_\{S\}\(C\)=\\emptysetand𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)⊈g\\mathit\{Extent\}\_\{S\}\(C\)\\nsubseteq gthen∃C∗=\(X,Y\)∈𝒜∗\\exists C^\{\*\}=\(X,Y\)\\in\\mathcal\{A\}^\{\*\}as𝐸𝑥𝑡𝑒𝑛𝑡S​\(C∗\)=𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)∖g≠∅\\mathit\{Extent\}\_\{S\}\(C^\{\*\}\)=\\mathit\{Extent\}\_\{S\}\(C\)\\setminus g\\neq\\emptyset\. This corresponds to the concepts whose extent containsg, whose simplified intent is empty and whose simplified extent is not contained byg\. The conceptCCremains asC∗C^\{\*\}with the same simplified intent, the simplified extent ofC∗C^\{\*\}may contain fewer objects if𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)∩g≠∅\\mathit\{Extent\}\_\{S\}\(C\)\\cap g\\neq\\emptyset\. 3. \(c\)If𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)=∅\\mathit\{Intent\}\_\{S\}\(C\)=\\emptysetand𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)⊆g\\mathit\{Extent\}\_\{S\}\(C\)\\subseteq gthen there does not exist any conceptC∗C^\{\*\}of𝒜∗\\mathcal\{A\}^\{\*\}such thatC∼C∗C\\sim C^\{\*\}\. In this case,𝐶𝐴\\mathit\{CA\}is not equivalent to any existing concept of𝒜\\mathcal\{A\}, otherwise the condition𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)⊆g\\mathit\{Extent\}\_\{S\}\(C\)\\subseteq gcould not be verified\. Thus,𝐶𝐴\\mathit\{CA\}is a new concept in𝒜∗\\mathcal\{A\}^\{\*\}, and a subconcept of the concept lattice\(X,Y\)\(X,Y\)that has no equivalent in𝒜∗\\mathcal\{A\}^\{\*\}\. In other words, the conceptCCwhose extent containsg, whose simplified extent is contained bygand whose simplified intent is empty has no equivalent in𝒜∗\\mathcal\{A\}^\{\*\}as the objects it introduces in𝒜\\mathcal\{A\}are introduced byCAin𝒜∗\\mathcal\{A\}^\{\*\}\. In Fig\.[18](https://arxiv.org/html/2609.00054#A1.F18)the barredCxC\_\{x\}is here to emphasize that no concept in𝒜\\mathcal\{A\}more specific thanCcan contain the objects introduced byC\.
4. 4\.ifg∩X≠∅g\\cap X\\neq\\emptysetandg⊈Xg\\nsubseteq XandX⊈gX\\nsubseteq g\. This corresponds to the concepts whose extent neither contains nor is contained bygbut shares some objects withg\. 1. \(a\)If𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)≠∅\\mathit\{Intent\}\_\{S\}\(C\)\\neq\\emptysetthen there existsC∗=\(X,Y\)C^\{\*\}=\(X,Y\)in𝒜∗\\mathcal\{A\}^\{\*\}\. If the simplified intent is not empty then an equivalent concept remains in𝒜∗\\mathcal\{A\}^\{\*\}\. Note that if some objects of the simplified extent are ing, then a conceptC2C\_\{2\}as pictured in dashed lines in Fig\.[18](https://arxiv.org/html/2609.00054#A1.F18)appears in𝒜∗\\mathcal\{A\}^\{\*\}introducing the objects from𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)∩g\\mathit\{Extent\}\_\{S\}\(C\)\\cap g\. 2. \(b\)If𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)=∅\\mathit\{Intent\}\_\{S\}\(C\)=\\emptysetand𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)⊈g\\mathit\{Extent\}\_\{S\}\(C\)\\nsubseteq gthen there existsC∗=\(X,Y\)C^\{\*\}=\(X,Y\)in𝒜∗\\mathcal\{A\}^\{\*\}\. If the simplified intent is empty but the simplified extent is not included ingthen the concept will remain in𝒜∗\\mathcal\{A\}^\{\*\}\. Note that if some of the simplified extent is included ing, then a conceptC2C\_\{2\}as pictured in dashed lines in Fig\.[18](https://arxiv.org/html/2609.00054#A1.F18)appears in𝒜∗\\mathcal\{A\}^\{\*\}introducing the objects from𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)∩g\\mathit\{Extent\}\_\{S\}\(C\)\\cap g\. 3. \(c\)If𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)=∅\\mathit\{Intent\}\_\{S\}\(C\)=\\emptysetand𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)⊂g\\mathit\{Extent\}\_\{S\}\(C\)\\subset gthen there does not exist any conceptC∗C^\{\*\}of𝒜∗\\mathcal\{A\}^\{\*\}such thatC∼C∗C\\sim C^\{\*\}\. LetC2C\_\{2\}be a concept from𝒦∗\\mathcal\{K\}^\{\*\}such that𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C2\)=g∩X\\mathit\{Extent\}\(C\_\{2\}\)=g\\cap X\(g∩Xg\\cap Xis closed in𝒦∗\\mathcal\{K\}^\{\*\}, as\(Y∪\{𝑎𝑡𝑡\}\)′=Y′∩g=X∩g\(Y\\cup\\\{\\mathit\{att\}\\\}\)^\{\\prime\}=Y^\{\\prime\}\\cap g=X\\cap g\)\.C2C\_\{2\}is a concept of𝒜∗\\mathcal\{A\}^\{\*\}asg∩𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)≠∅g\\cap\\mathit\{Extent\}\_\{S\}\(C\)\\neq\\emptysetandg∩𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)⊆𝐸𝑥𝑡𝑒𝑛𝑡S​\(C2\)g\\cap\\mathit\{Extent\}\_\{S\}\(C\)\\subseteq\\mathit\{Extent\}\_\{S\}\(C\_\{2\}\)\. There is noC3∈𝒜C\_\{3\}\\in\\mathcal\{A\}such thatC3∼C2C\_\{3\}\\sim C\_\{2\}\. In other words, if the simplified intent is empty and the whole simplified extent is included ingg, then the objects of the simplified extent are introduced by a conceptC2C\_\{2\}more specific thanCCleadingCCto disappear in𝒜∗\\mathcal\{A\}^\{\*\}\. In Fig\.[18](https://arxiv.org/html/2609.00054#A1.F18)the barredCxC\_\{x\}is here to emphasize that no concept in𝒜\\mathcal\{A\}more specific thanCcan contain the object introduced byC\.
5. 5\.ifg∩X=∅g\\cap X=\\emptyset,∃C∗∈𝒜∗\\exists C^\{\*\}\\in\\mathcal\{A\}^\{\*\}such thatC∼C∗C\\sim C^\{\*\}\. If the concept shares nothing withggthen it is unaffected by the addition ofatt\.

From the previous cases, only cases[3c](https://arxiv.org/html/2609.00054#A1.I1.i3.I1.i3)and[4c](https://arxiv.org/html/2609.00054#A1.I1.i4.I1.i3)can lead to removing a concept\. In all the other casesCChas an equivalent concept in𝒜∗\\mathcal\{A\}^\{\*\}, and the possible new concepts𝐶𝐴\\mathit\{CA\}can only enlarge𝐸𝑥𝑡𝒜∗\\mathit\{Ext\}\_\{\\mathcal\{A\}^\{\*\}\}\. However in a context of identified objects, cases[3c](https://arxiv.org/html/2609.00054#A1.I1.i3.I1.i3)and[4c](https://arxiv.org/html/2609.00054#A1.I1.i4.I1.i3)never occur: in both cases𝐼𝑛𝑡𝑒𝑛𝑡S​\(C\)=∅\\mathit\{Intent\}\_\{S\}\(C\)=\\emptyset\. SinceCCbelongs to the AOC\-poset, it introduces at least one element, here an objectoo, so thato∈𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)≠∅o\\in\\mathit\{Extent\}\_\{S\}\(C\)\\neq\\emptyset\. In a context of identified objects,\{o\}′′=\{o\}\\\{o\\\}^\{\\prime\\prime\}=\\\{o\\\}\. The object\-concept ofoois\(\{o\},\{o\}′\)\(\\\{o\\\},\\\{o\\\}^\{\\prime\}\), and sinceCCintroducesoo,CCis this object\-concept\. Therefore𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)=𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)=\{o\}\\mathit\{Extent\}\(C\)=\\mathit\{Extent\}\_\{S\}\(C\)=\\\{o\\\}and\|𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)\|=1\|\\mathit\{Extent\}\(C\)\|=1\. In[3c](https://arxiv.org/html/2609.00054#A1.I1.i3.I1.i3),g⊂𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)g\\subset\\mathit\{Extent\}\(C\)which means that\|g\|=0\|g\|=0, and𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)⊆g\\mathit\{Extent\}\_\{S\}\(C\)\\subseteq gwhich is impossible considering that\|𝐸𝑥𝑡𝑒𝑛𝑡S​\(C\)\|=1\|\\mathit\{Extent\}\_\{S\}\(C\)\|=1and\|g\|=0\|g\|=0\. In[4c](https://arxiv.org/html/2609.00054#A1.I1.i4.I1.i3),g∩𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)≠∅g\\cap\\mathit\{Extent\}\(C\)\\neq\\emptysetand𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)⊈g\\mathit\{Extent\}\(C\)\\nsubseteq gwhich is impossible considering that\|𝐸𝑥𝑡𝑒𝑛𝑡⁡\(C\)\|=1\|\\mathit\{Extent\}\(C\)\|=1\. Consequently, no concept of𝒜\\mathcal\{A\}is removed, and the AOC\-poset of a context of identified objects can only grow when an attribute is added\.

∎

## Appendix BAlgorithm RCA\-AOC\-conv

Algorithm[1](https://arxiv.org/html/2609.00054#alg1)implements the approach described in Sect\.[5\.4](https://arxiv.org/html/2609.00054#S5.SS4)\. Part of the notation used was introduced in Sect\.[2](https://arxiv.org/html/2609.00054#S2)and[3](https://arxiv.org/html/2609.00054#S3)\. We recall them for the sake of clarity and define the new ones\. We denote by:

- 1\.𝒫ip\\mathcal\{P\}\_\{i\}^\{p\}the concept\-poset built at stepppfor the context derived from𝒦i\\mathcal\{K\}\_\{i\}
- 2\.AOC​\-​poset​\(𝒦\)\\mathrm\{AOC\\text\{\-\}poset\(\\mathcal\{K\}\)\}the function used to compute the AOC\-poset of a context\. It is used at every step of the process
- 3\.trt\_\{r\}the index of the target context of relationrr
- 4\.RGiR\_\{G\_\{i\}\}the subset of relationsr∈𝐑r\\in\\mathbf\{R\}such thatGsr=GiG\_\{s\_\{r\}\}=G\_\{i\}, wheresrs\_\{r\}is the index of the source context of relationrr
- 5\.μ⁡\(m\)=\(\{m\}′,\{m\}′′\)\\mu\(m\)=\(\\\{m\\\}^\{\\prime\},\\\{m\\\}^\{\\prime\\prime\}\)the attribute\-concept ofmm, where the derivation operators are those of the current extended context𝒦ip\\mathcal\{K\}^\{p\}\_\{i\}\(used in the implementation note below\)
- 6\.γ⁡\(o\)=\(\{o\}′′,\{o\}′\)\\gamma\(o\)=\(\\\{o\\\}^\{\\prime\\prime\},\\\{o\\\}^\{\\prime\}\)the object\-concept ofoo, with the same convention

Algorithm[1](https://arxiv.org/html/2609.00054#alg1)makes the convergence argument directly visible: line[11](https://arxiv.org/html/2609.00054#alg1.l11)only ever adds relational attributes, whose incidence is fixed at creation and which are identified by their triple \(Definition[10](https://arxiv.org/html/2609.00054#Thmdefinition10)\), so that𝒦ip−1⊆𝒦ip\\mathcal\{K\}\_\{i\}^\{p\-1\}\\subseteq\\mathcal\{K\}\_\{i\}^\{p\}at every step and the number of attributes is bounded for each context\. The increasing sequence of extended contexts is therefore stationary \(Lemma[1](https://arxiv.org/html/2609.00054#Thmtheorem1)\) and the stop condition of the loop is eventually satisfied\. Note that scaling is applied to all the concepts of the previous posets: the attributes built on concepts already present at earlier steps exist in𝒦ip−1\\mathcal\{K\}\_\{i\}^\{p\-1\}and are not duplicated, while the attributes built at previous steps on concepts that have since been removed are kept by the apposition, as dangling attributes \(Sect\.[5\.4](https://arxiv.org/html/2609.00054#S5.SS4)\)\.

Algorithm 1RCA\-AOC\-conv1:an RCF

\(𝐊,𝐑\)\(\\mathbf\{K\},\\mathbf\{R\}\), with

𝐊\\mathbf\{K\}a set of formal contexts

\{𝒦i\}i=1,…,n\\\{\\mathcal\{K\}\_\{i\}\\\}\_\{i=1,\\dots,n\}, and a function

ρ\\rhowhich associates a scaling operator to each relation

2:a family of concept\-posets

\(𝒫i\)i=1,…,n\(\\mathcal\{P\}\_\{i\}\)\_\{i=1,\\dots,n\}
3:

p←0p\\leftarrow 0
4:for all

𝒦i∈𝐊\\mathcal\{K\}\_\{i\}\\in\\mathbf\{K\}do

5:

𝒦i0←𝒦i\\mathcal\{K\}\_\{i\}^\{0\}\\leftarrow\\mathcal\{K\}\_\{i\};

𝒫i0←AOC​\-​poset​\(𝒦i0\)\\mathcal\{P\}\_\{i\}^\{0\}\\leftarrow\\mathrm\{AOC\\text\{\-\}poset\}\(\\mathcal\{K\}\_\{i\}^\{0\}\)⊳\\trianglerightinitial AOC\-poset for𝒦i\\mathcal\{K\}\_\{i\}

6:endfor

7:repeat

8:

p←p\+1p\\leftarrow p\+1
9:for all

𝒦i∈𝐊\\mathcal\{K\}\_\{i\}\\in\\mathbf\{K\}do

10:

𝒞trp−1←𝑐𝑜𝑛𝑐𝑒𝑝𝑡​𝑠𝑒𝑡​𝑜𝑓​𝒫trp−1\\mathcal\{C\}\_\{t\_\{r\}\}^\{\\,p\-1\}\\leftarrow\\mathit\{concept\\penalty\\ set\\penalty\\ of\\penalty\\ \}\\mathcal\{P\}\_\{t\_\{r\}\}^\{\\,p\-1\}for each

r∈RGir\\in R\_\{G\_\{i\}\}⊳\\trianglerightwithRGi=\{r1i,…,rkii\}R\_\{G\_\{i\}\}=\\\{r^\{i\}\_\{1\},\\ldots,r^\{i\}\_\{k\_\{i\}\}\\\}

11:

𝒦ip←𝒦ip−1\|𝕊ρ⁡\(r1i\)​\(𝒦i,r1i,𝒞tr1ip−1\)​\|…\|​𝕊ρ⁡\(rkii\)​\(𝒦i,rkii,𝒞trkiip−1\)\\mathcal\{K\}\_\{i\}^\{p\}\\leftarrow\\mathcal\{K\}\_\{i\}^\{p\-1\}\\,\|\\,\\mathbb\{S\}\_\{\\rho\(r^\{i\}\_\{1\}\)\}\(\\mathcal\{K\}\_\{i\},r^\{i\}\_\{1\},\\mathcal\{C\}\_\{t\_\{r^\{i\}\_\{1\}\}\}^\{\\,p\-1\}\)\\,\|\\,\\dots\\,\|\\,\\mathbb\{S\}\_\{\\rho\(r^\{i\}\_\{k\_\{i\}\}\)\}\(\\mathcal\{K\}\_\{i\},r^\{i\}\_\{k\_\{i\}\},\\mathcal\{C\}\_\{t\_\{r^\{i\}\_\{k\_\{i\}\}\}\}^\{\\,p\-1\}\)⊳\\trianglerightapposition up to attribute identity \(Def\.[10](https://arxiv.org/html/2609.00054#Thmdefinition10)\): attributes already in𝒦ip−1\\mathcal\{K\}\_\{i\}^\{p\-1\}are not duplicated

12:

𝒫ip←AOC​\-​poset​\(𝒦ip\)\\mathcal\{P\}\_\{i\}^\{p\}\\leftarrow\\mathrm\{AOC\\text\{\-\}poset\}\(\\mathcal\{K\}\_\{i\}^\{p\}\)
13:endfor

14:until

∀i,𝒦ip=𝒦ip−1\\forall i,\\ \\mathcal\{K\}\_\{i\}^\{p\}=\\mathcal\{K\}\_\{i\}^\{p\-1\}
15:return

\(𝒫ip\)i=1,…,n\(\\mathcal\{P\}\_\{i\}^\{p\}\)\_\{i=1,\\dots,n\}

An implementation need not recompute the AOC\-poset from scratch at each step\. The one provided infca4jproceeds incrementally: \(i\) only the relational attributes referring to the concepts created at the previous step are generated, the others being already present; \(ii\) the new introducer concepts are obtained as the attribute\-conceptsμ⁡\(m\)\\mu\(m\)of the new attributesmm, and as the object\-conceptsγ⁡\(o\)\\gamma\(o\), recomputed for the objects owning a new attribute, since the introducer concept of an object may become more specific when new attributes are added; \(iii\) conversely, a concept whose introduced objects have all migrated to more specific concepts, and which introduces no attribute, is removed from the poset\. The incremental result coincides withAOC​\-​poset​\(𝒦ip\)\\mathrm\{AOC\\text\{\-\}poset\}\(\\mathcal\{K\}\_\{i\}^\{p\}\): the extent\{m\}′\\\{m\\\}^\{\\prime\}of the attribute\-concept of an existing attributemmis fixed and remains closed, and the object\-concept of an object owning no new attribute is unchanged\. This is the counterpart, for RCA\-AOC\-conv, of the optimization used in Galicia for RCA \(Sect\.[5\.4](https://arxiv.org/html/2609.00054#S5.SS4)\)\.

Table 12:RCF making RCA\-AOC diverge when using the existential scaling operator on each relation, and requiring object\-concept recomputation in RCA\-AOC\-conv\.Formal ContextsK1K\_\{1\}a1\_1o1\_1×\\timeso1\_2×\\timeso1\_3K2K\_\{2\}a2\_1a2\_2o2\_1×\\timeso2\_2×\\timesK3K\_\{3\}a3\_1a3\_2o3\_1×\\timeso3\_2×\\timesK4K\_\{4\}a4\_1a4\_2o4\_1×\\timeso4\_2×\\timesRelationsr​1​\_​2r1\\\_2o2\_1o2\_2o1\_1×\\timeso1\_2×\\timeso1\_3×\\timesr​3​\_​1r3\\\_1o1\_1o1\_2o1\_3o3\_1×\\timeso3\_2×\\timesr​4​\_​3r4\\\_3o3\_1o3\_2o4\_1×\\timeso4\_2×\\timesr​3​\_​4r3\\\_4o4\_1o4\_2o3\_1×\\timeso3\_2×\\times

We now develop an example which combines divergence and the need for recomputing object\-concepts when new attributes arrive\. Table[12](https://arxiv.org/html/2609.00054#A2.T12)shows the RCF\. The dependency graph is shown in Fig\.[19](https://arxiv.org/html/2609.00054#A2.F19)\. Figures[20](https://arxiv.org/html/2609.00054#A2.F20)and[21](https://arxiv.org/html/2609.00054#A2.F21)show the concept\-posets computed byfca4jwith RCA\-AOC\-conv at steps 0 to 4\.

![Refer to caption](https://arxiv.org/html/2609.00054v1/fig__dependency-graph-div-oc.png)Figure 19:Dependency graph of Table[12](https://arxiv.org/html/2609.00054#A2.T12)Figure 20:AOC\-posets built with RCA\-AOC\-conv at steps 0 to 2 for Table[12](https://arxiv.org/html/2609.00054#A2.T12)\. The process converges\.Figure 21:AOC\-posets built with RCA\-AOC\-conv at steps 3 to 4 for Table[12](https://arxiv.org/html/2609.00054#A2.T12)\. The process converges\.At step 0 \(Fig\.[20](https://arxiv.org/html/2609.00054#A2.F20), top\),K2K\_\{2\},K3K\_\{3\}andK4K\_\{4\}are contexts of identified objects\. InK1K\_\{1\},C\_K1\_2=\(\{o​1​\_​1,o​1​\_​2\},\{a​1​\_​1\}\)=\(\\\{o1\\\_1,o1\\\_2\\\},\\\{a1\\\_1\\\}\)introduceso1\_1ando1\_2, while the top conceptC\_K1\_3introduceso1\_3; inK3K\_\{3\},C\_K3\_6introduceso3\_1andC\_K3\_7introduceso3\_2; inK4K\_\{4\},C\_K4\_8introduceso4\_1andC\_K4\_9introduceso4\_2\.

At step 1 \(Fig\.[20](https://arxiv.org/html/2609.00054#A2.F20), second row\), two noteworthy concepts appear\. First,C\_K1\_10=\(\{o​1​\_​1\},\{a​1​\_​1,∃r​1​\_​2​\(C\_K2\_4\)\}\)=\(\\\{o1\\\_1\\\},\\\{a1\\\_1,\\exists r1\\\_2\(\\texttt\{C\\\_K2\\\_4\}\)\\\}\)is the new object\-concept ofo1\_1: it introduces no attribute \(a1\_1is introduced byC\_K1\_2and∃r​1​\_​2​\(C\_K2\_4\)\\exists r1\\\_2\(\\penalty\\texttt\{C\\\_K2\\\_4\}\)byC\_K1\_12\) and is only obtained by recomputing the object\-concept ofo1\_1, whose closure is refined by the new attribute \(point \(ii\) of the implementation note above\)\. Second,C\_K3\_13=\(\{o​3​\_​1,o​3​\_​2\}CLOSE,=\(\\\{o3\\\_1,o3\\\_2\\\},OPEN\{∃r​3​\_​1​\(C\_K1\_3\)\}\)\\\{\\exists r3\\\_1\(\\texttt\{C\\\_K1\\\_3\}\)\\\}\)has an empty simplified extent and refers toC\_K1\_3, which no longer belongs to the poset ofK1K\_\{1\}at this step: aso1\_3is now introduced byC\_K1\_12,C\_K1\_3no longer introduces any element and is removed, while the attribute∃r​3​\_​1​\(C\_K1\_3\)\\exists r3\\\_1\(\\texttt\{C\\\_K1\\\_3\}\), whose incidence is fixed \(Definition[10](https://arxiv.org/html/2609.00054#Thmdefinition10)\), is kept: it is our first example of a dangling attribute\. Note that the extent\{o​1​\_​1,o​1​\_​2,o​1​\_​3\}\\\{o1\\\_1,o1\\\_2,o1\\\_3\\\}ofC\_K1\_3belongs toE​x​t𝒫10Ext\_\{\\mathcal\{P\}\_\{1\}^\{0\}\}but not toE​x​t𝒫11Ext\_\{\\mathcal\{P\}\_\{1\}^\{1\}\}: RCA\-AOC\-conv does not preserve the step\-to\-step inclusion of the structures \(Sect\.[5\.4](https://arxiv.org/html/2609.00054#S5.SS4)\)\. This dangling reference will also be the seed of the divergence of RCA\-AOC\.

At step 2 \(Fig\.[20](https://arxiv.org/html/2609.00054#A2.F20), third row\), the presence ofC\_K1\_11andC\_K1\_12refines the intents of the object\-concepts ofK3K\_\{3\}:C\_K3\_6introduceso3\_1and gains∃r​3​\_​1​\(C\_K1\_11\)\\exists r3\\\_1\(\\texttt\{C\\\_K1\\\_11\}\), andC\_K3\_7introduceso3\_2and gains∃r​3​\_​1​\(C\_K1\_12\)\\exists r3\\\_1\(\\texttt\{C\\\_K1\\\_12\}\)\. The attribute∃r​3​\_​1​\(C\_K1\_10\)\\exists r3\\\_1\(\\texttt\{C\\\_K1\\\_10\}\), owned by no object, is introduced byC\_K3\_14, whose extent is empty\. In parallel,C\_K3\_13propagates into the circuit:C\_K4\_15=\(\{o​4​\_​1,o​4​\_​2\}CLOSE,=\(\\\{o4\\\_1,o4\\\_2\\\},OPEN\{∃r​4​\_​3​\(C\_K3\_13\)\}\)\\\{\\exists r4\\\_3\(\\penalty\\texttt\{C\\\_K3\\\_13\}\)\\\}\)is created, with an empty simplified extent as well\.

At step 3 \(Fig\.[21](https://arxiv.org/html/2609.00054#A2.F21), first row\), the convergence mechanism of RCA\-AOC\-conv becomes visible\. SinceC\_K4\_15persists, becauseC\_K3\_13also persists, the new attribute∃r​3​\_​4​\(C\_K4\_15\)\\exists r3\\\_4\(\\penalty\\texttt\{C\\\_K4\\\_15\}\)is simply added to the intent of the existing conceptC\_K3\_13, whose extent\{o​3​\_​1,o​3​\_​2\}\\\{o3\\\_1,o3\\\_2\\\}is already present: no concept is created inK3K\_\{3\}, and the mutual dependency betweenK3K\_\{3\}andK4K\_\{4\}is resolved instead of oscillating\. Similarly,C\_K4\_16introduces∃r​4​\_​3​\(C\_K3\_14\)\\exists r4\\\_3\(\\texttt\{C\\\_K3\\\_14\}\)with an empty extent\. At step 4 \(Fig\.[21](https://arxiv.org/html/2609.00054#A2.F21), second row\), the only change is the addition of∃r​3​\_​4​\(C\_K4\_16\)\\exists r3\\\_4\(\\texttt\{C\\\_K4\\\_16\}\)to the intent ofC\_K3\_14: the sets of extents are unchanged, the posets of steps 3 and 4 are equivalent \(Definition[9](https://arxiv.org/html/2609.00054#Thmdefinition9)\), and the algorithm stops at step 5\. In the final result,∃r​3​\_​1​\(C\_K1\_3\)\\exists r3\\\_1\(\\texttt\{C\\\_K1\\\_3\}\)remains a dangling attribute, in the intent ofC\_K3\_13: it is interpreted through the extent\{o​1​\_​1,o​1​\_​2,o​1​\_​3\}\\\{o1\\\_1,o1\\\_2,o1\\\_3\\\}of the conceptC\_K1\_3it was originally built on\.

Figure[22](https://arxiv.org/html/2609.00054#A2.F22)shows the concept\-posets computed byfca4jwith RCA\-AOC at steps 2 to 5, and the divergence of the process\. The first two steps are identical and not shown on the figure\. At step 2,C\_K3\_13disappears, sinceC\_K1\_3does not exist at step 1 and no other attribute is shared byo3\_1ando3\_2;C\_K4\_15then disappears at step 3, while an equivalent ofC\_K3\_13reappears \(C\_K3\_16\), built on the step 2 conceptC\_K4\_15\. The concepts on the circuit appear alternately, as in the counterexample of Table[9](https://arxiv.org/html/2609.00054#S4.T9): no two successive steps are equivalent, and the process loops with period two \(steps 3 and 5 share the same extents, as do steps 4 and 6\)\.

Note also that a variant of RCA\-AOC\-conv that would only compute the attribute\-concepts of the new attributes, skipping the recomputation of object\-concepts \(point \(ii\) of the implementation note above\), would converge as well, but would never createC\_K1\_10\.

Figure 22:AOC\-posets built with RCA\-AOC at steps 2 to 5 for Table[12](https://arxiv.org/html/2609.00054#A2.T12)\. The process diverges\.