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CMJV09I08P0914.Pdf Journal of Computer and Mathematical Sciences, Vol.9(8), 914-920 August 2018 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319 - 8133 (Online) Vague Filter of BL- Algebras S. Yahya Mohamed1 and P. Umamaheswari 2 1, 2Assistant Professors, PG and Research Department of Mathematics, Government Arts College, Tiruchirappalli, Tamil Nadu, INDIA. email: 1 [email protected]; [email protected]. (Received on: July 19, 2018) ABSTRACT In this paper, we introduce the notion of vague filter of BL- algebra, and investigate some important properties with illustrations. Further, we obtain some conditions of vague filter of BL- algebra. Mathematical Subject classification 2010: 03B47, 03G25, 03G10, 03E72, 06D99. Keywords: BL-algebra; Filter; Implicative filter; Vague set; Vague filter. 1. INTRODUCTION The concept of BL-algebras was introduced by Hảjek8 as the algebraic structure for his Basic Logic. A well known example of BL- algebra is the interval [0, 1] endowed with the structure induced by a continuous t- norm. On other hand, the MV- algebras, introduce by Chang2 in 1958 are one of the most well known classes of BL-algebras. In order to investigate the logic system whose semantic truth value is given by a lattice, Xu11 proposed the concept of lattice implication algebras. Xu and Qin12 investigated the properties of filters of lattice implication algebras. Zadeh13 in 1965 introduced the notion of fuzzy set to describe vagueness mathematically in its very abstractness and tried to solve such problems by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. These membership grades are very often represented by real number ranging in the closed interval 0 and 1. Fuzzy set idea has been applied to other algebraic structures such as groups, semi groups, rings, modules, vector spaces, and topologies and widely used in many fields. Meanwhile, the deficiency of fuzzy set is a single function, and it cannot express the evidence of supporting and opposing. For this reason, the concept of vague set was introduced in 1993 914 S. Yahya Mohamed, et al., Comp. & Math. Sci. Vol.9 (8), 914-920 (2018) by Gau and Buehrer7. The idea of vague set is the same with the idea of intuitionistic fuzzy set1, 2; so, the vague set is equivalent to intuitionistic fuzzy set. In the present paper, we introduce the notion of a vague filter in BL- algebra, and investigate some related properties. Moreover, we give some conditions for a vague set to be a vague filter. 2. PRELIMINARIES In this section, we recall some basic definitions and their properties which are helpful to develop the main results. Definition 2.1[8] A BL-algebra is an algebra (퐴, ∨, ∧, ∗, →, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (i) (퐴, ∨, ∧, 0, 1) is a bounded lattice, (ii) (퐴, ∗, 1)is a commutative monoid, (iii) ∗ and → form an adjoint pair, that is, 푧 ≤ 푥 → 푦 if and only if 푥 ∗ 푧 ≤ 푦 for all 푥, 푦, 푧 ∈ 퐴, (iv) 푥 ∧ 푦 = 푥 ∗ (푥 → 푦), (v) (푥 → 푦) ∨ (푦 → 푥) = 1. Proposition 2.2[9] In a BL- algebra A, the following properties hold for all 푥, 푦, 푧 ∈ 퐴, (i) 푦 → (푥 → 푧) = 푥 → (푦 → 푧) = (푥 ∗ 푦) → 푧, (ii) 1 → 푥 = 푥, (iii) 푥 ≤ 푦 if and only if 푥 → 푦 = 1, (iv) 푥 ∨ y = ((푥 → 푦) → 푦) ∧ ((푦 → 푥) → 푥), (v) 푥 ≤ 푦 implies 푦 → 푧 ≤ 푥 → 푧, (vi) 푥 ≤ 푦 implies 푧 → 푥 ≤ 푧 → 푦, (vii) 푥 → 푦 ≤ (푧 → 푥) → (푧 → 푦), (viii) 푥 → 푦 ≤ (푦 → 푧) → (푥 → 푧), (ix) 푥 ≤ (푥 → 푦) → 푦, (x) 푥 ∗ (푥 → 푦) = 푥 ∧ y, (xi) 푥 ∗ 푦 ≤ 푥 ∧ 푦 (xii) 푥 → 푦 ≤ (푥 ∗ 푧) → (푦 ∗ 푧), (xiii) 푥 ∗ (푦 → 푧) ≤ 푦 → (푥 ∗ 푧), (xiv) (푥 → 푦) ∗ (푦 → 푧) ≤ 푥 → 푧, − (xv) (푥 ∗ 푥 ) = 0. Note. In the sequel, we shall use 퐴 to denote as BL- algebras and the operation ∨, ∧, ∗ have priority towards the operations " → ". Note. In a BL- algebra 퐴, we can define 푥− = 푥 → 0 for all 푥 ∈ 퐴. 915 S. Yahya Mohamed, et al., Comp. & Math. Sci. Vol.9 (8), 914-920 (2018) Definition 2.3[14] A filter of a BL- algebra 퐴 is a nonempty subset F of 퐴 such that for all 푥, 푦 ∈ 퐴, (i) If 푥, 푦 ∈ 퐹, then 푥 ∗ 푦 ∈ 퐹, (ii) If 푥 ∈ 퐹 and 푥 ≤ 푦, then 푦 ∈ 퐹. Proposition 2.4[14] Let 퐹 be a nonempty subset of a BL- algebra A. Then F is a filter of 퐴 if and only if the following conditions hold (i) 1 ∈ 퐹, (ii) 푥, 푥 → 푦 ∈ 퐹 implies 푦 ∈ 퐹. A filter F of a BL-algebra A is proper if 퐹 ≠ 퐴. Definition 2.5[3, 6, 7] A vague set 푆 in the universe of discourse 푋 is characterized by two membership functions given by (i) A truth membership function 푡푆: 푋 → [0, 1], (ii) A false membership function 푓푆: 푋 → [0, 1]. Where 푡푆(푥) is lower bound of the grade of membership of x derived from the ‘evidence for x’, and 푓푆(푥) is a lower bound of the negation of x derived from the ‘evidence against x’ and 푡푆(푥)+푓푆(푥) ≤ 1. Thus the grade of membership of x in the vague set S is bounded by a subinterval [푡푆(푥), 1 − 푓푆(푥)] of [0, 1]. The vague set 푆 is written as 푆 = {(푥, [ 푡푆(푥), 푓푆(푥)])/푥 ∈ 푋}, where the interval [푡푆(푥), 1 − 푓푆(푥)] is called the value of x in the vague set 푆 and denoted by 푉푆(푥). Definition 2.6[7] A vague set 푆 of a set 푋 is called (i) the zero vague set of 푋 if 푡푆(푥) = 0 and 푓푆(푥) = 1 for all 푥 ∈ 푋, (ii) the unit vague set of 푋 if 푡푆(푥) = 1 and 푓푆(푥) = 0 for all 푥 ∈ 푋, (iii) the 훼- vague set of 푋 if 푡푆(푥) = 훼 and 푓푆(푥) = 1 − 훼 for all 푥 ∈ 푋, where 훼 ∈ (0, 1). Definition 2.7[7] Let 푆 be a vague set of 푋 with truth membership function 푡푆 and the false membership function 푓푆. For any 훼, 훽 ∈ [0, 1], the (훼, 훽)-cut of the vague set 푋 is crisp subset 푆(훼,훽)of the set 푋 by 푆(훼,훽) = {푉(푥) ≥ [훼, 훽]/푥 ∈ 푋}. Obviously, 푆(0,0) = 푋. Definition 2.8[7] Let 퐷[0, 1] denote the family of all closed subintervals of [0, 1]. Now we define refined maximum (rmax) and “≥ " on elements 퐷1 = [푎1, 푏1] and 퐷2[푎2, 푏2] of 퐷[0, 1] as 푟푚푎푥 (퐷1, 퐷2) = [max{푎1, 푎2} , max{푏1, 푏2}]. Similarly we can define ≤, = and rmin. 3. PROPERTIES OF VAGUE FILTER In this section, we introduce the notion of vague filter of BL- algebra and investigate some relevant properties with illustrations. Definition 3.1 Let 푆 be vague set of a BL-algebra 퐴 is called a vague filter of 퐴 if it satisfies the following axioms (i) 푉푆(1) ≥ 푉푆(푥), (ii) 푉푆(푦) ≥ 푟푚푖푛{푉푆(푥 → 푦), 푉푆(푥)} for all 푥, 푦 ∈ 퐴. 916 S. Yahya Mohamed, et al., Comp. & Math. Sci. Vol.9 (8), 914-920 (2018) Proposition 3.2 Let 푆 be vague set of BL-algebra 퐴 is a vague filter of 퐴 if and only if the following hold if for any 푥, 푦 ∈ 퐴, (i) 푡푆(1) ≥ 푡푆(푥) and 1 − 푓푆(1) ≥ 1 − 푓푆(푥), (ii) 푡푆(푦) ≥ min{푡푆(푥 → 푦), 푡푆(푥)} and 1 − 푓푆(푦) ≥ min {1 − 푓푆(푥 → 푦), 1 − 푓푆(푦)}. Proof Let 푆 be vague filter of 퐴. Then from (i) of definition 3.1 and the definition of 푉푆, we have (i) straight forward. From (ii) of definition 3.1and the definition of 푉푆, (ii) is obvious.□ The following is the example of definition 3.1 and proposition 3.2. Example 3.3 Let 퐴 = {0, 푎, 푏, 1}. The binary operations ∗ and → give by the following tables (1) and (2): Table (1) Table (2) ∗ 0 a b 1 → 0 a b 1 0 0 0 0 0 0 1 1 1 1 a 0 0 a b a a 1 1 1 b 0 a 1 1 b 0 a b b 1 0 a b 1 1 0 a b 1 Then (퐴, ∨, ∧, ∗, →, 0, 1) is a BL- algebra. Define a vague set 푆 of 퐴 as follows: 푆 = {(1, [0.7, 0.2]), (푎, [0.5, 0.3]), (푏, [0.5, 0.3]), (0, [0.7, 0.2])}. It is easily verify that 푆 is vague filter of A and satisfy the conditions (i) and (ii) of proposition 3.2. Proposition 3.4 Every vague filter 푆 of BL- algebra 퐴 is order preserving. Proof Let 푆 be vague filter of BL-algebra 퐴. Then we prove that if for any 푥, 푦 ∈ 퐴, 푥 ≤ 푦 then 푉푆(푥) ≤ 푉푆(푦). From (ii) of proposition 3.2, we have, 푡푆(푦) ≥ min{푡푆(푥 → 푦), 푡푆(푥)} = min {푡푆(1), 푡푆(푥)}, [From (iii) of proposition 2.2] Also, we have 1 − 푓푆(푦) ≥ min{1 − 푓푆(푥 → 푦), 1 − 푓푆(푦)}. From (i) of proposition 3.2, we have 푡푆(1) ≥ 푡푆(푥) and 1 − 푓푆(1) ≥ 1 − 푓푆(푥).
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