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In this paper,we first introduce vague set,then give the definition of intersection and union based on vague set with t-norm and t-conorm of the point value. Thus we gain some properties of intersection and union of vague set.

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The concepts of degree of truth-compatibility, degree of false-compatibility degree of truth-equality, degree of falscequality based on t-norm and t-conorm were introduced, because a vague set has the characteristic of truth-membership function and fase-membership function. Futhermore we presented the concepts of semi-vague partitions and vague partitions by using locically degree of truth-compatibility, degree of false-compatibility degree of truth-equality,degree of falscequality, and we investigated the characters of semi-vague partitions and vague partitions.

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By using vague sets we generalize the notion of convex sets and introduce the notion of (α, β ,T )-convex vague sets and study their properties, where T is a triangular norm on [0, 1].

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The purpose of this paper is to construct topology on vague soft sets. The concept of vague soft topology is introduced and its basic properties are given.

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Based on the presented theory of Vague sets and the topological theory of classical sets and Fuzzy sets,this paper spreaded the relative topological theory of classical sets and Fuzzy sets through the method of analysis, presented the basic concepts of Vague topological space and Vague continuous mapping, and discussed some relative properties of them. These results further extended the scope of the research on Vague sets, and proposed the future study aspects in this field.

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In this note, by using the concept of vague sets, the notion of vague BCK/BCI-algebra is introduced. And the notions of $\alpha$-cut and vague-cut are introduced and the relationships between these notions and crisp subalgebras are studied.

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Vague set is a valid tool for processing uncertain information.The similarity measure of two uncertain patterns is important for intelligent reasoning.It is also a key problem to measure the similarity of vague values or vague sets in vague information processing systems.Many methods for similarity measure of vague sets have been proposed in recent years.However,these methods could not precisely describe the essences of the similarity between two vague sets.In this paper,some evaluation criterions for simil...

Abstract:
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