Catálogo de publicaciones - libros

The Galois Sub-hierarchy (GSH) is a polynomial-size representation of a concept lattice which has been applied to several fields, such as software engineering and linguistics.

In this paper, we analyze the performances, in terms of computation time, of three GSH-building algorithms with very different algorithmic strategies: , and . We use Java and C++ as implementation languages and Galicia as our development platform.

Our results show that implementations in C++ are significantly faster, and that in most cases Pluton is the best algorithm.

In [1] a generalisation of Formal Concept Analysis was introduced with data mining applications in mind, -Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the standard Boolean carrier set. A fundamental result was missing there, namely the second half of the equivalent of the main theorem of Formal Concept Analysis. In this continuation we introduce the structural lattice of such generalised contexts, providing a limited equivalent to the main theorem of -Formal Concept Analysis which allows to interpret the standard version as a privileged case in yet another direction. We motivate our results by providing instances of their use to analyse the confusion matrices of multiple-input multiple-output classifiers.

Several fuzzifications of formal concept analysis have been proposed to deal with uncertainty or incomplete information. In this paper, we focus on the new paradigm of multi-adjoint concept lattices which embeds different fuzzy extensions of concept lattices, our main result being the representation theorem of this paradigm. As a consequence of this theorem, the representation theorems of the other paradigms can be proved more directly. Moreover, the multi-adjoint paradigm enriches the language providing greater flexibility to the user.

We study the “non-unit implications” of a formal context and investigate the closure system induced by these implications. It turns out that this closure system is the largest closure system on the same base set containing the given one as a complete sublattice. This was studied by other authors with special emphasis on semidistributivity and convex geometries. We present some of their results in FCA language.

The complete lattice refinements of a closure system form an interval within the lattice of all closure systems. We describe the reduced context for this interval.

For better compatibility with the literature, we dualize and consider implications between objects, not attributes.

In the paper we study lattices of axiomatizable classes and relatively axiomatizable classes. This study is based on Formal Concept Analysis [4,5]. The notion of a relatively axiomatizable class is a generalization of such concepts as variety, quasivariety, ∀-axiomatizable class, -axiomatizable class, -axiomatizable class, -axiomatizable class and so on. Relatively axiomatizable classes were studied in [10]. It is proved in the paper that any finite lattice may be represented as the lattice of all relatively axiomatizable subclasses of the class of all models of a one-element signature with respect to some set of sentences. Also we prove that any finite or countable complete lattice is isomorphic to the lattice of all relatively axiomatizable subclasses of some class of models with respect to a proper set of sentences.

Double Boolean algebras were introduced in [Wi00a] as a variety fundamental for Boolean Concept Logic, an extension of Formal Concept Analysis allowing negations of formal concepts. In this paper, the free double Boolean algebra generated by the constants is described. Moreover, we show that every free double Boolean algebra with at least one generator is infinite. A measure of the complexity of terms specific for double Boolean algebras is introduced. This, together with a modification of the algorithm for protoconcept exploration (cf. [Vo04]) yields double Boolean algebras containing a counterexample to every term identity up to a given complexity if the identity does not hold in general. These algebras can be constructed automatically, thus the word problem for free double Boolean algebras is solved.

The MacNeille completion of a poset (, ≤ ) is the smallest (up to isomorphism) complete poset containing (, ≤ ) that preserves existing joins and existing meets. It is wellknown that the MacNeille completion of a Boolean algebra is a Boolean algebra. It is also wellknown that the MacNeille completion of a distributive lattice is not always a distributive lattice (see [Fu44]). The MacNeille completion even seems to destroy many properties of the initial lattice (see [Ha93]). Weakly dicomplemented lattices are bounded lattices equipped with two unary operations satisfying the equations (1) to (3’) of Theorem 3. They generalise Boolean algebras (see [Kw04]). The main result of this contribution states that the MacNeille completion of a weakly dicomplemented lattice is a weakly dicomplemented lattice. The needed definitions are given in subsections 1.2 and 1.3.

06B23.

The following paper activates polynomial methods for data analysis.We want to embed a given formal context into a polynomial context ) in such a way that implications can be computed in the polynomial context, using the algebraic structure. The basic ideas of formal concept analysis that are needed here are sketched in [TB1].

This paper offers a mathematical analysis of labelled line diagrams of finite concept lattices to gain a better understanding of those diagrams. The main result is the . This Theorem can be applied to justify, for instance, the training tool “ ” which has been created to support the understanding and the drawing of appropriate line diagrams of finite concept lattices.

There exists a close relation between the Ferrers-dimension of a context and the order dimension of the appropriate concept lattice [4]. Based on this fact we will introduce on contexts and show how they characterize planar concept lattices.