Multi-Fuzzy Extensions of Functions

Fuzzy Sets and Systems, 2003

The relations between fuzzy preorders, fuzzy consequence operators, fuzzy closure systems and co-closure systems lattice structures are studied. Given a continuous t-norm * , we will prove that the complete lattice of all fuzzy * -preorders is isomorphic with a subfamily of the complete lattice of fuzzy consequence operators. Moreover, we will show that it is possible to embed both structures in the class of fuzzy closure systems by a lattice dual monomorphism between semilattices. A solution to the following open problem: "when two families of fuzzy possible valuations produce the same fuzzy implication relation" is given. Concept of * -interior operator of a fuzzy consequence operator is suggested and a characterization of the previous open problem through * -interior operator is shown.

2005

In this paper a theory of relations and corelations between the lattice of fuzzy subsets of a crisp set X and that of a crisp set Y is developed, based on the theory of relations and corelations between textures. In a series of examples it is shown that these notions generalize in a natural way the impotant concept of fuzzy relation from X to Y. Difunctions are also characterized and their relationship with known mappings between fuzzy sets is investigated.

Iranian Journal of Fuzzy Systems, 2011

The starting point of this paper is given by Priestley’s papers, where a theory of representation of distributive lattices is presented. The purpose of this paper is to develop a representation theory of fuzzy distributive lattices in the finite case. In this way, some results of Priestley’s papers are extended. In the main theorem, we show that the category of finite fuzzy Priestley spaces is equivalent to the dual of the category of finite fuzzy distributive lattices. Several examples are also presented.

Kybernetika, 2021

In this paper, we introduce the product, coproduct, equalizer and coequalizer notions on the category of fuzzy implications on a bounded lattice that results in the existence of the limit, pullback, colimit and pushout. Also isomorphism, monic and epic are introduced in this category. Then a subcategory of this category, called the skeleton, is studied. Where none of any two fuzzy implications are Φ-conjugate.

Fuzzy Sets and Systems, 2001

For crisp sets X and Y , considering two fuzzy equalities EX ; EY on X and Y , respectively, the concept of fuzzy function and several types of it were studied in M. Demirci, Fuzzy Sets and Systems 106 (1999) 239 -246. In this paper, for a fuzzy relation f on X × Y , considering the fuzzy equalities EX on X and EY on Y the degree of f of being fuzzy function, being surjective, being injective and being bijective is deÿned. Then setting the connections between several types of degree of being fuzzy function and various types of fuzzy function the essential properties of these degrees are investigated.

International Mathematical Forum, 2008

In this paper we present two theorems that rely on the Zadeh's extension principle. These two theorems can be used to define a crisp function on a given fuzzy real number. And this will produce a new fuzzy real number. Using this, we can define some special fuzzy numbers such as: square root, natural logarithm, logarithm,... etc.

The problem of embedding incomplete into complete relations has been an important topic of research in the context of crisp relations. After Szpilrajn’s result, several variations have been published. Alcantud studied in Economic Theory (2009) the case where the extension is asked to satisfy some order conditions between elements. He first studied and solved a particular formulation where conditions are imposed in terms of strict preference only, which helps to precisely identify which quasiorderings can be extended when we allow for additional conditions in terms of indifference too. In this contribution we generalize both results to the fuzzy case.

The class of overtaker binary relations a ssociated with the order in a lattice is dened and used to generalize the r epresentations of L-fuzzy sets by m eans of level sets or fuzzy points.

UPB Scientific Bulletin, Series A: Applied Mathematics and Physics

Inspired by the study of algebraic structures of soft sets, our aim in this paper is to initiate research on the connection between soft sets and lattices. We apply the notion of soft sets to lattices. Some related notions are defined and several basic properties are discussed by using the soft set theory. Furthermore, the concept of ∈-soft set, q-soft set and q k -soft set is introduced and some interesting properties are investigated. Using the notion of ”belongingness” and ”quasicoincidence” of fuzzy points and fuzzy sets, characterizations for an ∈-soft set, q-soft set and q k -soft set to be a soft ideal (filter) are established.

In this paper we present a method to construct fuzzy-valued t-norms and t-conorms, i.e. operations which map pairs of lattice elements to fuzzy sets, and are commutative, associative and monotone. The fuzzy-valued t-norm and t-conorm are synthesized from their α-cuts which are obtained from families of multi-valued t-norms and t-conorms.