I. Mardones-Pérez

Fuzzy Galois connections categorically

This paper presents a systematic investigation of fuzzy (non-commutative) Galois connections in the sense of R. Bělohlavek [1], from the point of view of enriched category theory. The results obtained show that the theory of enriched categories makes it possible to present the theory of fuzzy Galois connections in a succinct way; and more importantly, it provides a useful method to express and to study the link and the difference between the commutative and the non-commutative worlds (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A representation theorem for fuzzy pseudometrics

In this paper, we show that there exists a one to one correspondence between a certain class of fuzzy pseudometrics (in the sense of Kramosil and Michalek) and [0,1)-indexed families of ordinary pseudometrics satisfying a property of lower semicontinuity. The aforementioned bijection is proved to be independent of the t-norm and it provides a representation theorem for a large class of fuzzy pseudometric spaces. Further, the relations between the uniformities and topologies both generated by the fuzzy pseudometric and by the corresponding family of ordinary pseudometrics are also investigated.

Higher separation axioms in L -topologically generated I( L )-topological spaces

The main result of this paper is a theorem about inserting a pair of semicontinuous L-real-valued functions which extends the insertion theorem of Kubiak [Comment. Math. Univ. Carolinae 34 (1993) 357-362] from L = {0, 1} to an arbitrary meet-continuous lattice L (endowed with an order-reversing involution). With this result it is shown that the normality-type separation axioms in TOP(L) are preserved by the functor which takes an L-topological space X to the I(L)-topological space ΩL(X) obtained by providing the set X with the I(L)-topology consisting of all lower semicontinuous functions from X to I(L). The same is proved for the case of the regularity axiom.

Uniform-type structures on lattice-valued spaces and frames

By introducing lattice-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L an integral commutative quantale). By showing, via an intermediate L-valued structure of uniformity, how filters of covers may describe the uniform operators of Hutton, we prove that, when restricted to Girard quantales, this general framework captures a significant class of Huttons uniform spaces. The categories of L-valued uniform spaces and L-valued uniform frames here introduced provide (in the case L is a complete chain) the missing vertices in the commutative cube formed by the classical categories of topological and uniform spaces and their corresponding pointfree counterparts (forming the base of the cube) and the corresponding L-valued categories (forming the top of the cube).

Fuzzy pseudometric spaces vs fuzzifying structures

In this paper, we apply the representation theorem established in Mardones-Perez and de Prada Vicente (2012) 12 to define and study the degree in which some topological-type properties of fuzzy pseudometric spaces are fulfilled. Fuzzifying structures which appear naturally are also investigated, and the relation between these structures and fuzzy pseudometric spaces is explored.

An insertion theorem for continuous -valued functions and its consequences

Spaces in which open sets take values in the L-interval I(L) are investigated. We prove a theorem concerning the insertion of a continuous function with values in I(L) with L a complete lattice. We establish certain factorizations of the functors ωI(L) and ιI(L) with L a hypercontinuous lattice. The latter result and the insertion theorem provide a partial answer to a recent open question related to insertion of lattice-valued functions. As another application of the insertion theorem we establish a relation between complete regularity in the categories of L-topological and I(L)-topological spaces with L a frame.

An application of a representation theorem for fuzzy metrics to domain theory

In this paper we study the interplay between fuzzy metric spaces and domain theory using the representation theorem established in Mardones-Perez and de Prada Vicente (2012) 12. We will prove that suitable restrictions of completeness in fuzzy metric spaces correspond to a structure of domain in the poset of formal balls endowed with appropriate orders.

On the uniformization of lattice-valued frames

In this note we discuss the appropriate way of uniformizing the notion of a lattice-valued frame introduced by Pultr and Rodabaugh in 2003. We cover the case of a completely distributive lattice (which is, in a certain sense, the most general one) and study the corresponding category of uniform lattice-valued frames. In particular, we show that this is a complete and cocomplete category that extends in a nice manner the category of uniform frames, widely studied in the literature.