Operadores adjuntos en entornos generales y sus aplicaciones

Abstract

Adjoint triples arise as a generalization of a t-norm and its residuated implication. They are basic operators to make the calculus in multi-adjoint logic programming, multi-adjoint concept lattices, multi-adjoint fuzzy rough sets and multi-adjoint fuzzy relation equations, providing more flexibility and increasing the range of applications in the setting in which they are considered.

This thesis is focused on the study of the adjoint triples, their properties and applications. Firstly, important properties of adjoint pairs/triples and the algebraic structures associated with these operators, which are called multi-adjoint algebras, have been presented. Later, this work presents an intense comparison among di erent general algebraic structures such as implication triples, sup-preserving aggregations, quantales, u-norms, uninorms and general implications considered in extended-order algebras. This comparative study proves that the use of these algebraic structures, in environments requiring residuated implications, provides particular cases of multi-adjoint algebras.

Moreover, adjoint negations are introduced as a new generalization of residuated negations that satisfy the most signifi cant properties. Besides generalizing this kind of negations, this work shows that adjoint negations generalize, at least, three of the most useful negation operators given in the literature, such as the negation operators introduced by Trillas, the pairs of weak negations presented by Georgescu and Popescu and the negation operators de fined by Della Stella and Guido in the setting of a specifi c extended-order algebra.

The last part of the thesis describes a process on how to represent a multi-adjoint logic programming as a multi-adjoint relation equation, which is important in order to use these fuzzy relation equations as a decision support system for fuzzy logic. The solvability of these equations is provided from the theory of Fuzzy Formal Concept Analysis.