Congruencias y factorización como herramientas de reducción en el análisis de conceptos formales

Resumo

Since its introduction at the beginning of the eighties by B. Ganter and R. Wille, Formal Concept Analysis (FCA) has been one of the most developed mathematical tools for data analysis. FCA is a mathematical theory that determines conceptual structures among datasets. In particular, the databases considered in this theory are called contexts and are composed of a set of objects, a set of attributes and a relationship between the sets. The tools provided by FCA can properly manipulate data and extract relevant information from it. One of the most relevant and intensively developed research lines is the reduction of the set of attributes contained in these datasets, preserving the original information and removing the redundancy it may contain. Attribute reduction has also been studied in other environments, such as in Rough Set Theory, as well as in the different fuzzy generalizations of FCA and Rough Set Theory. On the one hand, it has been shown that when an attribute reduction is carried out in a formal context, an equivalence relation is induced on the set of concepts obtained from the original context. This induced equivalence relation has a particularity, its equivalence classes have a join semilattice structure with a maximum element, i.e., in general, they could not have a closed algebraic structure. In this thesis, we study how it is possible to enhance the attribute reduction by endowing equivalence classes with a closed algebraic structure. The notion of congruence achieves this purpose, however, the use of this kind of equivalence relation may cause a great loss of information due to the fact that the generated equivalence classes group many concepts. In order to address this issue, in this thesis a weakened notion of congruence is introduced and it is called local congruence. The local congruence gives rise equivalence classes with the structure of convex sublattice, being more flexible when it groups concepts but maintaining interesting properties from an algebraic point of view.