Inma P. Cabrera

Fuzzy Logic, Soft Computing, and Applications

We survey on the theoretical and practical developments of the theory of fuzzy logic and soft computing. Specifically, we briefly review the history and main milestones of fuzzy logic (in the wide sense), the more recent development of soft computing, and finalise by presenting a panoramic view of applications: from the most abstract to the most practical ones.

On galois connections and soft computing

After recalling the different interpretations usually assigned to the term Galois connection, both in the crisp and in the fuzzy case, we survey on several of their applications in Computer Science and specifically, in Soft Computing.

On the definition of suitable orderings to generate adjunctions over an unstructured codomain

Given a mappingf:A->B from a (pre-)ordered set A into an unstructured set B, we study the problem of defining a suitable (pre-)ordering relation on B such that there exists a mapping g:B->A such that the pair of mappings (f,g) forms an adjunction between (pre-)ordered sets. The necessary and sufficient conditions obtained are then expressed in terms of closure operators and closure systems.

On residuation in multilattices: Filters, congruences, and homomorphisms

Continuing with our general study of algebraic hyperstructures, we focus on the residuated operation in the framework of multilattices. Firstly, we recall the existing relation between filters, homomorphisms and congruences in the framework of multilattices; then, introduce the notion of residuated multilattice and further study the notion of filter, which has to be suitably modified so that the results in the first section are conveniently preserved also in the residuated case.

Congruence relations on some hyperstructures

In this work we study the structure of the set of congruences on several hyperstructures with one and two (hyper-)operations. On the one hand, we show sufficient conditions guaranteeing that the set of congruences of an nd-groupoid forms a complete lattice (which, in turn, is a sublattice of the lattice of equivalence relations on the nd-groupoid). On the other hand, we focus on the study of the congruences on a multilattice; specifically, we prove that the set of congruences on an m-distributive multilattice forms a complete lattice and, moreover, show that the classical relationship between homomorphisms and congruences can be adequately adapted to work with multilattices under suitable restrictions.

A coalgebraic approach to non-determinism: Applications to multilattices

Multilattices are a suitable generalization of lattices which enables to accommodate the formalization of non-deterministic computation; specifically, the algebraic characterization for multilattices provides a formal framework to develop tools in several fields of computer science. On the other hand, the usefulness of coalgebra theory has been increasing in the recent years, and its importance is undeniable. In this paper, somehow mimicking the use of universal algebra, we define a new kind of coalgebras (the ND-coalgebras) that allows to formalize non-determinism, and show that several concepts, widely used in computer science are, indeed, ND-coalgebras. Within this formal context, we study a minimal set of properties which provides a coalgebraic definition of multilattices.

Generating Isotone Galois Connections on an Unstructured Codomain

Given a mapping f : A → B from a partially ordered set A into an unstructured set B, we study the problem of defining a suitable partial ordering relation on B such that there exists a mapping g : B → A such that the pair of mappings (f,g) forms an isotone Galois connection between partially ordered sets.

Fuzzy congruence relations on nd-groupoids

In this work we introduce the notion of fuzzy congruence relation on an nd-groupoid and study conditions on the nd-groupoid that guarantee a complete lattice structure on the set of fuzzy congruence relations. The study of these conditions allowed to construct a counterexample to the statement that the set of fuzzy congruences on a hypergroupoid is a complete lattice.

On the construction of adjunctions between a fuzzy preposet and an unstructured set

In this work, we focus on adjunctions, also called isotone Galois connections, in the framework of fuzzy preordered sets (hereafter, fuzzy preposets). Specifically, we present necessary and sufficient conditions so that, given a mapping f:A→B from a fuzzy preposet A into an unstructured set B, it is possible to construct a suitable fuzzy preorder relation on B for which there exists a mapping g:B→A such that the pair (f,g) constitutes an adjunction.

On the Existence of Isotone Galois Connections between Preorders

Given a mapping f : A → B from a preordered set A into an unstructured set B, we study the problem of defining a suitable preordering relation on B such that there exists a mapping g : B → A such that the pair (f,g) forms an adjunction between preordered sets.