FILTERS OF MTL-ALGEBRAS BASED ON VAGUE SET THEORY

Abstract

Abstract. In this paper, we introduce the concept of a vague lter ofMTL-algebra, and then some related properties are investigated. 1. IntroductionZadeh[13] introduced the concept of fuzzy as a new mathematical toolfor dealing with uncertainties, several researches were conducted on the gener-alization of the notion of fuzzy sets. The idea of \vague set was rst publishedby Gau and Buehrer [3], as a generalization of the notion of fuzzy set. Estevaand Godo[2] introduced a new algebra, called an MTL-algebra, and studiedseveral basic properties. MTL-algebras are algebraic structures for monoidalt-norm based logic (MTL), a many-valued propositional calculus that formal-izes the structure of the real interval [0;1];induced by a left-continuous t-norm.They also introduced the notion of lters in MTL-algebras. Zhang [12] stud-ied further properties of lters in MTL-algebras. Using the vague set, Biswas[1] studied vague groups. Jun and Park [6, 8] studied vague ideals and vaguedeductive systems in subtraction algebras. In this paper, we introduce thenotion of vague lters in MTL-algebras, and then some related properties areinvestigated.2. PreliminariesIn this section, we collect some de nition and results that have been used inthe sequel.De nition 1. ([4]) An algebra (L;;^;_; ;!;0;1) with four binaryoperation and two constant is a residuated lattice if it satis es:(R1) (L;;^;_;0;1) is a lattice with the least element 0 and the largestelement 1,(R2) is a commutative semigroup with the unit element 1,