Multi-adjoint Concept Lattices Research Articles

Attribute implications in multi-adjoint concept lattices with hedges

The computation of (if-then) rules relating the most representative variables of a given dataset is a relevant goal in different areas of data analysis such as stock market prediction, disease diagnosis and census data analysis, among others. In Formal Concept Analysis (FCA), attribute implications follow the philosophy of the if-then rules establishing relationships between sets of attributes. This paper will focus on the theoretical development of attribute implications through the tools provided by the multi-adjoint concept lattice framework. We will show different notions to compute the validity of attribute implications, analyzing their properties and comparing the proposed alternatives. The use of linguistic hedges, and truth-stressing hedges in particular, is an interesting way of incorporating truth functions into attribute implications. Moreover, they have been considered in the computation of the concepts of a given context. This paper also introduces and studies the main properties of the multi-adjoint concept lattices framework with truth-stressing hedges.

Characterizing One-Sided Formal Concept Analysis by Multi-Adjoint Concept Lattices

Managing and extracting information from databases is one of the main goals in several fields, as in Formal Concept Analysis (FCA). One-sided concept lattices and multi-adjoint concept lattices are two frameworks in FCA that have been developed in parallel. This paper shows that one-sided concept lattices are particular cases of multi-adjoint concept lattices. As a first consequence of this characterization, a new attribute reduction mechanism has been introduced in the one-side framework.

Multi-adjoint concept lattices via quantaloid-enriched categories

With quantaloids carefully constructed from multi-adjoint frames, it is shown that multi-adjoint concept lattices, multi-adjoint property-oriented concept lattices and multi-adjoint object-oriented concept lattices are derivable from Isbell adjunctions, Kan adjunctions and dual Kan adjunctions between quantaloid-enriched categories, respectively.

Attribute Classification and Reduct Computation in Multi-Adjoint Concept Lattices

The problem of reducing information in databases is an important topic in formal concept analysis, which has been studied in several articles. In this article, we consider the fuzzy environment of the multi-adjoint concept lattices, since it is a general fuzzy framework that allows us to easily establish degrees of preference on the elements of the considered database. We introduce algorithms to discover the information contained in the relational system. By means of these algorithms, we classify the attributes of a multi-adjoint context, and build a minimal subset of attributes preserving the information of the original knowledge system.

Characterizing reducts in multi-adjoint concept lattices

The construction of reducts, that is, minimal sets of attributes containing the main information of a database, is a fundamental task in different frameworks, such as in Formal Concept Analysis (FCA) and Rough Set Theory (RST). This paper will be focused on a general fuzzy extension of FCA, called multi-adjoint concept lattice, and we present a study about the attributes generating meet-irreducible elements and on the reducts in this framework. From this study, we introduce interesting results on the cardinality of reducts and the consequences in the classical case.

Attribute and size reduction mechanisms in multi-adjoint concept lattices

Attribute reduction and size reduction in concept lattices are key research topics in Formal Concept Analysis (FCA). This paper combines both strategies in the multi-adjoint concept lattice framework in order to simplify the information provided by the original context. Specifically, we present three procedures which merge the attribute reduction and the size reduction by means of an irreducible α-cut concept lattice, analyzing the obtained properties.

On the use of irreducible elements for reducing multi-adjoint concept lattices

Looking for strategies to reduce the size of concept lattices is very important in formal concept analysis, when they preserve the main information of the relational database.This paper presents several properties of the useful fuzzy-attributes, in the general fuzzy case of multi-adjoint concept lattices and provides two mechanisms in order to reduce the size of concept lattices based on irreducible elements, without losing or modifying important information. Specifically, the reduced concept lattices are sublattices of the original one. Moreover, interesting properties of these mechanisms are studied and the relationship among both and other strategies is also introduced.

Attribute reduction in multi-adjoint concept lattices

Knowledge reduction is one of the key issues in formal concept analysis and there have been many studies on this topic. The irreducible elements in a lattice are also very important, since they form the basic information of a relational system. Moreover, they are also important from the viewpoint of attribute reduction.Both topics are notably more complicated in a fuzzy setting since not only the size of the sets of attributes and objects influence the size of the fuzzy concept lattice, but the truth-value sets, where the sets of objects, attributes and the relation are evaluated, are important.This paper presents, in the general fuzzy framework of multi-adjoint concept lattices, a characterization of the meet-irreducible elements, from which a classification of attributes and its application to attribute reduction is introduced.

On the use of thresholds in multi-adjoint concept lattices

The size of the concept lattices increases exponentially from the number of objects and attributes. This situation is more complicated in the fuzzy case, in which the considered carriers to evaluate the objects and attributes, and for the relation, are also taken into account. Hence, it is very important to study mechanisms to reduce the size of fuzzy concept lattices maintaining the main information. One of the most important mechanisms to reduce the size of concept lattices is the use of thresholds in the concept-forming operators. This paper studies this mechanism in the general fuzzy framework of multi-adjoint concept lattices, obtaining interesting properties and consequences.

Multi-adjoint concept lattices with heterogeneous conjunctors and hedges

This paper is related, on the one hand, to the framework of multi-adjoint concept lattices with heterogeneous conjunctors and, on the other hand, to the use of intensifying hedges as truth-stressers. Specifically, we continue on the line of recent works by Belohlavek and Vychodil, which use intensifying hedges as a tool to reduce the size of a concept lattice. In this paper we use hedges as a reduction tool in the general framework of multi-adjoint concept lattices with heterogeneous conjunctors.

On basic conditions to generate multi-adjoint concept lattices via Galois connections

This paper introduces sufficient and necessary conditions with respect to the fuzzy operators considered in a multi-adjoint frame under which the standard combinations of multi-adjoint sufficiency, possibility, and necessity operators form (antitone or isotone) Galois connections. The underlying idea is to study the minimal algebraic requirements so that the concept-forming operators (defined using the same syntactical form than the extension and intension operators of multi-adjoint concept lattices) form a Galois connection. As a consequence, given a relational database, we have much more possibilities to construct concept lattices associated with it, so that we can choose the specific version which better suits the situation.

Dual multi-adjoint concept lattices

Several papers relate different alternative approaches to classical concept lattices: such as property-oriented and object-oriented concept lattices and the dual concept lattices. Whereas the usual approach to the latter is via a negation operator, this paper presents a fuzzy generalization of the dual concept lattice, the dual multi-adjoint concept lattice, in which the philosophy of the multi-adjoint paradigm is applied and no negation on the lattices is needed.

A comparative study of adjoint triples

Adjoint triples are helpful as basic operators used in several domains. For example, adjoint triples play an important role in two important frameworks: multi-adjoint logic programming and multi-adjoint concept lattices. This paper shows that adjoint triples are an interesting generalization of t-norms and their residuated implications, since they preserve their main properties as well as they help to increase the flexibility of the operators used for computation in the considered framework. Furthermore, these operators will be related to other important general triples, specifically implication triples, and the definition of these latter operators will be improved.

On multi-adjoint concept lattices based on heterogeneous conjunctors

Sets of attributes and objects in fuzzy formal concept analysis are usually different and, hence, it might not make sense to evaluate them on the same carrier. In this context, the operators used to obtain the concept lattice could be defined by associating different lattices to attributes and objects; several reasons exist for which we need to evaluate the sets of attributes and objects in the same carrier. Following this direction, we introduce a new definition of a concept lattice, where objects and attributes are evaluated on the same lattice L, although operators evaluating objects and attributes in different carriers are used. Moreover, we study the relationship between this new concept lattice and the alternative one which can be obtained directly by using different carriers for the sets of attributes and objects.

Multi-adjoint property-oriented and object-oriented concept lattices

This paper presents a generalisation of the classical property and object-oriented concept lattices to a fuzzy environment based on the philosophy of the multi-adjoint paradigm. These concept lattices are generalisations of rough set theory used to consider two different sets – the set of objects and the set of attributes – to apply the corresponding modal operators, as in formal concept analysis.First of all, the paper presents several specific properties of adjoint triples in which the duals of several supports are considered. These properties are then used to introduce the multi-adjoint property and object-oriented concept lattices, in which different adjoint triples can be used in non-linear sets, as well as the corresponding representation (fundamental) theorems. Moreover, these lattices are related to the multi-adjoint concept lattice, in which negation operators are not needed; as a result, this relation allows the properties to be translated from one to another.

Formal concept analysis via multi-adjoint concept lattices

Several fuzzifications of formal concept analysis have been proposed to deal with uncertain information. In this paper, we focus on concept lattices under a multi-adjoint paradigm, which enriches the language providing greater flexibility to the user in that he/she can choose from a number of different connectives. Multi-adjoint concept lattices are shown to embed different fuzzy extensions of concept lattices found in the literature, the main results of the paper being the representation theorem of this paradigm and the embedding of other well-known approaches.