Javier Gutiérrez García

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)

Unified Representability of Total Preorders and Interval Orders through a Single Function: The Lattice Approach

We introduce a new approach that deals, jointly and in a unified manner, with the topics of numerical (continuous) representability of total preorders and interval orders. This setting is based on the consideration of increasing scales and the systematic use of a particular kind of codomain, that has a key lattice theoretical structure of a completely distributive lattice and allows us to use a single function (taking values in that codomain) in order to represent both kinds of binary relations.

UNIVERSAL CODOMAINS TO REPRESENT INTERVAL ORDERS

Given a binary relation defined on a set, we study its representability by means of a monotonic function that takes values on a suitable universal codomain (that depends on the kind of relation considered). We pay an special attention to the representability of interval orders, studying their alternative universal codomains, some of them equivalent to the set of symmetric triangular fuzzy numbers.

Tensor products of complete lattices and their application in constructing quantales

Abstract This paper aims to implement tensor products of complete lattices into fuzzy set theory. The most convenient approach for this purpose is to view the tensor product of two complete lattices as the family of all join reversing maps between those lattices. We show that some fundamental constructions in fuzzy set theory are tensor products. Examples of such circumstances include the following (to mention only three of them): the complete lattice of all lower semicontinuous maps from a topological space into a continuous lattice is the tensor product of the topology of the space and the range lattice; a binary operation coming from Zadehs extension principle is the tensor product of the Minkowski multiplication with the multiplication of the underlying unital quantale; triangle functions on nonnegative left-continuous distribution functions are tensor products of the real unit interval and the extended nonnegative half-line equipped, respectively, with a left-continuous t -norm and the usual addition.

Numerical Representation of Semiorders

We introduce a codomain to represent semiorders, total preorders and interval orders by means of a single map. We characterize the semiorders that are representable in the extended real line.

On lattice-valued frames: The completely distributive case

We provide an extension of the notion of chain-valued frame introduced by Pultr and Rodabaugh in [Category theoretic aspects of chain-valued frames: parts I and II, Fuzzy Sets and Systems 159 (2008) 501-528 and 529-558] by relaxing the assumption that L be a complete chain. As a result of this investigation we formulate the category L-Frm of L-frames under the weaker assumption that L is a completely distributive lattice. In particular, L-Frm is complete and cocomplete. Finally we prove that, in a certain sense, the assumption of L being a completely distributive lattice cannot be weakened.

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).

Forty years of Hutton fuzzy unit interval

This note is a very short history of a specific fuzzy topological space – the Huttons fuzzy unit interval [0,1](L) – which occupies one of the central positions in fuzzy topology. On that occasion the superiority of L-valued topology over [0,1]-valued topology is being discussed. We then show how the Huttons concept evolved. After obvious identifications we can today say that [0,1](L) is the tensor product [0,1]⊗L of the lattices [0,1] and L. An account of the consequences of this new view point is hoped to appear thereafter.

Fuzzy Galois connections on fuzzy sets

Abstract In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the readership of fuzzy set theorists, with particular attention paid to fuzzy Galois connections between preordered fuzzy sets.

On fuzzy uniformities induced by a fuzzy metric space

Abstract Different types of fuzzy uniformities have been introduced in the literature standing out the notions due to Hutton, Hohle and Lowen. The main purpose of this paper is to study several methods to endow a fuzzy metric space ( X , M , ⁎ ) , in the sense of George and Veeramani, with a probabilistic uniformity and with a Hutton [ 0 , 1 ] (-quasi)-uniformity. We will show the functorial behavior of these constructions as well as its relation with respect to Lowens functors and Katsarass functors, which establish a relationship between the categories of probabilistic uniformities and Hutton [ 0 , 1 ] (-quasi)-uniformities with the category of classical uniformities respectively. Furthermore, we also study the fuzzy topologies associated with these fuzzy uniformities.