Vague Filters of Residuated Lattices

Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice and we find its distributive sublattices. The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters. 1. Introduction In the classical set, there are only two possibilities for any elements: in or not in the set. Hence the values of elements in a set are only one of and . Therefore, this theory cannot handle the data with ambiguity and uncertainty. Zadeh introduced fuzzy set theory in 1965 [1] to handle such ambiguity and uncertainty by generalizing the notion of membership in a set. In a fuzzy set each element is associated with a point-value selected from the unit interval , which is termed the grade of membership in the set. This membership degree contains the evidences for both supporting and opposing . A number of generalizations of Zadeh’s fuzzy set theory are intuitionistic fuzzy theory, L-fuzzy theory, and vague theory. Gau and Buehrer proposed the concept of vague set in 1993 [2], by replacing the value of an element in a set with a subinterval of . Namely, a true membership function and a false-membership function are used to describe the boundaries of membership degree. These two boundaries form a subinterval of . The vague set theory improves description of the objective real world, becoming a promising tool to deal with inexact, uncertain, or vague knowledge. Many researchers have applied this theory to many situations, such as fuzzy control, decision-making, knowledge discovery, and fault diagnosis. Recently in [3], Jun and Park introduced the notion of vague ideal in pseudo MV-algebras and Broumand Saeid [4] introduced the notion of vague BCK/BCI-algebras. The concept of residuated lattices was introduced by Ward and Dilworth [5] as a generalization of the structure of the set of ideals of a ring. These algebras are a common structure among algebras associated with logical systems (see [6–9]). The residuated lattices have interesting algebraic and logical properties. The main example of residuated lattices related to logic is and BL-algebras. A basic logic algebra (BL-algebra for short) is an important class of logical algebras introduced by Hajek [10] in