Ivan Mezzomo

On fuzzy ideals of fuzzy lattice

We characterize a fuzzy lattice through a fuzzy partial order relation, define a fuzzy ideal and fuzzy filter of fuzzy lattice, characterize a fuzzy ideal of fuzzy lattice using its level set and its support and show that a subset of a fuzzy lattice is a fuzzy ideal if and only if its support is a crisp ideal. Similarly, we show the same for its level set.

Operations on bounded fuzzy lattices

We consider the notion of fuzzy lattice introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374), and define the operations of product and collapsed sum on bounded fuzzy lattice analogous to the classical theory. Also, we prove that the product and collapsed sum on bounded fuzzy lattices are fuzzy posets and, consequently, bounded fuzzy lattices.

α-ideals of fuzzy lattices

We consider the fuzzy lattice notion introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374), define an α-ideals and α-filters for fuzzy lattices and characterize α-ideals and α-filters of fuzzy lattices by using its support and its level set. Moreover, we prove some similar properties to the classical theory of α-ideals and α-filters, such as, the class of α-ideals and α-filters are closed under union and intersection.

On n-dimensional strict fuzzy negations

n-dimensional fuzzy sets are an extension of fuzzy sets where the membership values are n-truples of real numbers in the unit interval [0, 1] ordered in increasing order, called n-dimensional intervals. The set of n-dimensional intervals is denoted by Ln([0, 1]). This paper aims to investigate the class of functions on Ln([0, 1]) which are continuous and strictly decreasing, called n-dimensional strict fuzzy negations. In particular, investigate the class of representable n-dimensional strict fuzzy negations, i.e., n-dimensional strict fuzzy negation which are determined by strict fuzzy negation. The main properties of strict fuzzy negations on [0, 1] are preserved by representable strict fuzzy negations on Ln([0, 1]). In addition, the conjugate obtained by action of an n-dimensional automorphism on an n-dimensional strict fuzzy negation provides a method to obtain other n-dimensional strict fuzzy negations, in which the properties of the original one are preserved, as well as the Fodors characterization theorem.

Types of fuzzy ideals in fuzzy lattices

In this paper we consider the notion of Fuzzy Lattices, which was introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374). We propose some new notions for Fuzzy Ideals and Filters and provide a characterization of Fuzzy Ideals via α-level Sets and Support. Some types of ideals and filters, such as: Fuzzy Principal Ideals (Filters), Proper Fuzzy Ideals (Filters), Prime Fuzzy Ideals (Filters) and Fuzzy Maximal Ideals (Filters) are also provided. Some properties (analogous to the classical theory) are also proved and the notion of Homomorphism from fuzzy lattices as well as the demonstration of some important propositions about it are also provided.

Fuzzy a-Ideals of Product Operator on Bounded Fuzzy Lattices

We consider the fuzzy lattice notion introduced by Chon, characterize a fuzzy ideal on operation of product between bounded fuzzy lattices. Define fuzzy a-ideals of fuzzy lattices and some properties analogous to the classical theory are also proved. Moreover, we characterize a fuzzy a-ideal on operation of product between bounded fuzzy lattices and prove results involving a fuzzy a-ideal of the product operator between fuzzy lattices and the product between fuzzy α-ideals of the bounded fuzzy lattices.

Natural n-dimensional fuzzy negations for n-dimensional t-norms and t-conorms

N-dimensional fuzzy sets are an extension of fuzzy sets where the membership values are n-tuples of real numbers in the unit interval [0,1] ordered in increasing order, called n-dimensional intervals. The set of n-dimensional intervals is denoted by Ln([0,1]). In the present paper, we consider the definitions and results obtained for n-dimensional fuzzy negations, applying these studies mainly on natural n-dimensional fuzzy negations for n-dimensional triangular norms and triangular conorms. Additionally, the conjugate obtained by action of an n-dimensional automorphism on an n-dimensional natural fuzzy negations for n-dimensional triangular norms and triangular conorms, provides a method to obtain other n-dimensional strong fuzzy negations, in which its properties on Ln([0,1]) are preserved.

Fuzzy α-ideals of collapsed sum operator on bounded fuzzy lattices

We consider the notion of fuzzy lattices introduced by Chon and characterize fuzzy ideals in terms of the collapsed sum operator between two bounded fuzzy lattices L and M. We also define fuzzy a-ideals in fuzzy lattices and demonstrate the relation between fuzzy a-ideals of the collapsed sum on bounded fuzzy lattices.

Equilibrium Point of Representable Moore Continuous n-Dimensional Interval Fuzzy Negations

n-dimensional interval fuzzy sets are a type of fuzzy sets which consider ordered n-tuples in \([0,1]^n\) as membership degree. This paper considers the notion of representable n-dimensional interval fuzzy negations, in particular, these that are Moore continuous, proposed in a previous paper of the authors, and we study some conditions that guarantee the existence of equilibrium point in classes of representable (Moore continuous) n-dimensional interval fuzzy negations. In addition, we prove that the changing of the dimensions of representable Moore continuous n-dimensional fuzzy negations inherits their equilibrium points.

A class of fuzzy implications obtained from triples of fuzzy implications

The aim of this work is to study the class I<inf>I, I1, I2</inf>(U) of fuzzy implications obtained by a triple (I, I<inf>1</inf>, I<inf>2</inf>) of fuzzy implications. Thus, this paper discusses under which conditions such functions preserve the main properties of fuzzy implications. In addition, by conjugate fuzzy implications it is shown that an I<inf>I, I1, I2</inf>(U) fuzzy implication can be preserved by action of an order automorphism. Finally, we introduce the family I^(k)<inf>I, I1, I2</inf>(U) of fuzzy implications obtained by taking the extended classes of I^(k)-implications verifying both generalized properties, exchange principle and distributivity in addition to, their dual construction is also considered.