Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 41, 2021, pp. 214-226 214 Properties of an anti-vague filter in BL- algebras S. Yahya Mohamed* P. Umamaheswari† Abstract In this paper, the concept of an anti-vague filter of a BL-algebra is introduced with suitable illustration, and also obtained some related properties. Further, we have investigated some more equivalent conditions of anti-vague filter. Keywords: BL-algebra; filter; implicative filter; vague set; vague filter; anti-vague filter 2010 AMS subject classification‡: 03B50; 03B52; 03E72; 06D35. *Assistant Professors, PG and Research Department of Mathematics, Government Arts College, Tiruchirappalli-620 022. Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India; yahya_md@yahoo.com †Assistant Professors, PG and Research Department of Mathematics, Government Arts College, Tiruchirappalli-620 022. Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India; umagactrichy@gmail.com ‡ Received on August 28, 2021. Accepted on November 13, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.650. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. Properties of an anti-vague filter in BL- algebras 215 1. Introduction Hảjek [5] introduced the idea of BL-algebras as the algebraic structure for his Basic Logic. The interval [0, 1] endowed with the structure induced by a continuous t- norm is a well-known example of BL- algebra. The MV- algebras, on the other hand, are one of the most well-known groups of BL- algebras, having been introduced by Chang [2] in 1958. In 1965, Zadeh [12] introduced the concept of a fuzzy set. The flaw in fuzzy sets is that they only have one feature, which means they cannot convey supporting and opposing data. Gau and Buehrer [4] introduced the principle of vague set in 1993 as a result of this. The authors [7, 8, 9, 10] discussed the vague filter, implicative filter, prime, and Boolean implicative filters of BL- algebras, as well as some of their properties. The frame work of this study is constructed as follow: some basic observations connected to anti-vague filter are provided in “Preliminaries”. “Anti-vague filter” presents the new notions of anti-vague filter in BL-algebra and investigated some related properties, also derived some equivalent conditions for an anti-vague filter to be a vague filter. Finally, the conclusion is presented in “Conclusion”. 2. Preliminaries In this section, we will go through some basic BL-algebra, filter, and vague set concepts, as well as their properties, which will help in the development of the main results. Definition 2.1[5] A BL-algebra is an algebra (𝐴, ∨, ∧, ∗, →, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (i) (𝐴, ∨, ∧, 0, 1) is a bounded lattice, (ii) (𝐴, ∗, 1)is a commutative monoid, (iii) ∗ and → form an adjoint pair, that is, 𝑧 ≤ 𝑥 → 𝑦 if and only if 𝑥 ∗ 𝑧 ≤ 𝑦 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (iv) 𝑥 ∧ 𝑦 = 𝑥 ∗ (𝑥 → 𝑦), (v) (𝑥 → 𝑦) ∨ (𝑦 → 𝑥) = 1. Proposition 2.2[6] In a BL- algebra A, the following properties are hold for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) 𝑦 → (𝑥 → 𝑧) = 𝑥 → (𝑦 → 𝑧) = (𝑥 ∗ 𝑦) → 𝑧, (ii) 1 → 𝑥 = 𝑥, (iii) 𝑥 ≤ 𝑦 if and only if 𝑥 → 𝑦 = 1, (iv) 𝑥 ∨ y = ((𝑥 → 𝑦) → 𝑦) ∧ ((𝑦 → 𝑥) → 𝑥), (v) 𝑥 ≤ 𝑦 implies 𝑦 → 𝑧 ≤ 𝑥 → 𝑧, S. Yahya Mohamed and P. Umamaheswari 216 (vi) 𝑥 ≤ 𝑦 implies 𝑧 → 𝑥 ≤ 𝑧 → 𝑦, (vii) 𝑥 → 𝑦 ≤ (𝑧 → 𝑥) → (𝑧 → 𝑦), (viii) 𝑥 → 𝑦 ≤ (𝑦 → 𝑧) → (𝑥 → 𝑧), (ix) 𝑥 ≤ (𝑥 → 𝑦) → 𝑦, (x) 𝑥 ∗ (𝑥 → 𝑦) = 𝑥 ∧ y, (xi) 𝑥 ∗ 𝑦 ≤ 𝑥 ∧ 𝑦 (xii) 𝑥 → 𝑦 ≤ (𝑥 ∗ 𝑧) → (𝑦 ∗ 𝑧), (xiii) 𝑥 ∗ (𝑦 → 𝑧) ≤ 𝑦 → (𝑥 ∗ 𝑧), (xiv) (𝑥 → 𝑦) ∗ (𝑦 → 𝑧) ≤ 𝑥 → 𝑧, (xv) (𝑥 ∗ 𝑥−) = 0. Note. In the sequel, we shall use 𝐴 to denote as BL- algebras and the operation ∨, ∧, ∗ have priority towards the operations " → ". Note. In a BL- algebra 𝐴, we can define 𝑥 − = 𝑥 → 0 for all 𝑥 ∈ 𝐴. Definition 2.3[13] A filter of a BL- algebra 𝐴 is a non-empty subset F of 𝐴 such that for all 𝑥, 𝑦 ∈ 𝐴, (i) If 𝑥, 𝑦 ∈ 𝐹, then 𝑥 ∗ 𝑦 ∈ 𝐹, (ii) If 𝑥 ∈ 𝐹 and 𝑥 ≤ 𝑦, then 𝑦 ∈ 𝐹. Proposition 2.4[13] Let 𝐹 be a non-empty subset of a BL- algebra A. Then, 𝐹 is a filter of 𝐴 if and only if the following conditions are hold (i) 1 ∈ 𝐹, (ii) 𝑥, 𝑥 → 𝑦 ∈ 𝐹 implies 𝑦 ∈ 𝐹. A filter F of a BL-algebra A is proper if 𝐹 ≠ 𝐴. Definition 2.5[1, 3, 4] A vague set 𝑆 in the universe of discourse 𝑋 is characterized by two membership functions given by (i) A truth membership function 𝑡𝑆 : 𝑋 → [0, 1], (ii) A false membership function 𝑓𝑆 : 𝑋 → [0, 1]. Where 𝑡𝑆 (𝑥) is lower bound of the grade of membership of x derived from the ‘evidence for x’, and 𝑓𝑆 (𝑥) is a lower bound of the negation of x derived from the ‘evidence against x’ and 𝑡𝑆 (𝑥)+𝑓𝑆 (𝑥) ≤ 1. Thus the grade of membership of x in the vague set S is bounded by a subinterval [𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)] of [0, 1]. The vague set 𝑆 is written as 𝑆 = {(𝑥, [ 𝑡𝑆 (𝑥), 𝑓𝑆 (𝑥)])/𝑥 ∈ 𝑋}, where the interval [𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)] is called the value of x in the vague set 𝑆 and denoted by 𝑉𝑆(𝑥). Definition 2.6[4] A vague set 𝑆 of a set 𝑋 is called (i) the zero vague set of 𝑋 if 𝑡𝑆 (𝑥) = 0 and 𝑓𝑆 (𝑥) = 1 for all 𝑥 ∈ 𝑋, (ii) the unit vague set of 𝑋 if 𝑡𝑆 (𝑥) = 1 and 𝑓𝑆 (𝑥) = 0 for all 𝑥 ∈ 𝑋, Properties of an anti-vague filter in BL- algebras 217 (iii) the 𝛼- vague set of 𝑋 if 𝑡𝑆 (𝑥) = 𝛼 and 𝑓𝑆 (𝑥) = 1 − 𝛼 for all 𝑥 ∈ 𝑋, where 𝛼 ∈ (0, 1). Definition 2.7[4] Let 𝑆 be a vague set of 𝑋 with truth membership function 𝑡𝑆 and the false membership function 𝑓𝑆. For all 𝛼, 𝛽 ∈ [0, 1], the (𝛼, 𝛽)-cut of the vague set 𝑋 is crisp subset 𝑆(𝛼,𝛽)of the set 𝑋 by 𝑆(𝛼,𝛽) = {𝑉(𝑥) ≥ [𝛼, 𝛽]/𝑥 ∈ 𝑋}. Obviously, 𝑆(0,0) = 𝑋. Definition 2.8[4] Let 𝐷[0, 1] denote the family of all closed subintervals of [0, 1]. Now, we define refined maximum (rmax) and “≥ " on elements 𝐷1 = [𝑎1, 𝑏1] and 𝐷2[𝑎2, 𝑏2] of 𝐷[0, 1] as 𝑟𝑚𝑎𝑥 (𝐷1, 𝐷2) = [max{𝑎1, 𝑎2} , max{𝑏1, 𝑏2}]. Similarly, we can define ≤, = and rmin. 3. Anti-Vague Filter In this section, we introduce the notion of an anti-vague filter of BL- algebra with illustration. Moreover, we discuss some related properties. Definition 3.1 Let 𝑆 be vague set of a BL-algebra 𝐴 is called an anti vague filter of 𝐴 if it satisfies the following axioms (i) 𝑉𝑆(1) ≤ 𝑉𝑆(𝑥), (ii) 𝑉𝑆(𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑥)} for all 𝑥, 𝑦 ∈ 𝐴. Proposition 3.2 Let 𝑆 be vague set of BL-algebra 𝐴. 𝑆 is an anti vague filter of 𝐴 if and only if the following hold if for all 𝑥, 𝑦 ∈ 𝐴, (i) 𝑡𝑆 (1) ≤ 𝑡𝑆 (𝑥) and 1 − 𝑓𝑆 (1) ≤ 1 − 𝑓𝑆 (𝑥), (ii) 𝑡𝑆 (𝑦) ≤ max{𝑡𝑆 (𝑥 → 𝑦), 𝑡𝑆 (𝑥)} and 1 − 𝑓𝑆 (𝑦) ≤ max {1 − 𝑓𝑆 (𝑥 → 𝑦), 1 − 𝑓𝑆 (𝑦)}. Proof: Let 𝑆 be an anti-vague filter of 𝐴. Then from (i) of definition 3.1 and the definition of 𝑉𝑆, we have (i) straight forward. From (ii) of definition 3.1and the definition of 𝑉𝑆, (ii) is obvious.∎ The following is the example of definition 3.1 and proposition 3.2. Example 3.3 Let 𝐴 = {0, 𝑎, 𝑏, 1}. The binary operations ′ ∗ ′ and ′ → ′ give by the following tables 3.1 and 3.2: S. Yahya Mohamed and P. Umamaheswari 218 Table3.1: ‘ ∗ ′Operator Table 3.2: ‘→’ Operator Then (𝐴, ∨, ∧, ∗, →, 0, 1) is a BL- algebra. Define a vague set 𝑆 of 𝐴 as follows: 𝑆 = {(1, [0.2, 0.7]), (𝑎, [0.3, 0.5]), (𝑏, [0.3, 0.5]), (0, [0.2, 0.7])}. It is easily verified that 𝑆 is an anti-vague filter of 𝐴 and satisfy the conditions (i) and (ii) of proposition 3.2. Proposition 3.4 Every anti-vague filter 𝑆 of BL- algebra 𝐴 is order preserving. Proof: Let 𝑆 be an anti-vague filter of BL-algebra 𝐴. Then, we prove that if 𝑥 ≤ 𝑦, then 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦) for all 𝑥, 𝑦 ∈ 𝐴. From (ii) of the proposition 3.2, we have, 𝑡𝑆 (𝑦) ≤ max{𝑡𝑆 (𝑥 → 𝑦), 𝑡𝑆 (𝑥)} ∗ 0 a b 1 0 0 0 0 0 a 0 0 a b b 0 a b b 1 0 a b 1 → 0 a b 1 0 1 1 1 1 a a 1 1 1 b 0 a 1 1 1 0 a b 1 Properties of an anti-vague filter in BL- algebras 219 = max {𝑡𝑆 (1), 𝑡𝑆 (𝑥)}, [From (iii) of proposition 2.2] Also, we have 1 − 𝑓𝑆 (𝑦) ≤ max{1 − 𝑓𝑆 (𝑥 → 𝑦), 1 − 𝑓𝑆 (𝑦)}. From (i) of the proposition 3.2, we have 𝑡𝑆 (1) ≤ 𝑡𝑆 (𝑥) and 1 − 𝑓𝑆 (1) ≤ 1 − 𝑓𝑆 (𝑥). Thus, 𝑡𝑆 (𝑦) ≤ max{𝑡𝑆(𝑥), 1 − 𝑓𝑆 (𝑦)} ≤ 1 − 𝑓𝑆 (𝑦), and so 𝑉𝑆 (𝑦) = [𝑡𝑆(𝑦), 1 − 𝑓𝑆 (𝑦)] ≤ [𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)] = 𝑉𝑆(𝑥). Hence 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦). ∎ Proposition 3.5 Let 𝑆 be a vague set of BL- algebra 𝐴, 𝑆 be an anti-vague filter of A if and only if 𝑥 → (𝑦 → 𝑧) = 1 implies 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴. Proof: Let 𝑆 be an anti-vague filter of BL-algebra 𝐴. Then, from (ii) of the definition 3.1, we have 𝑉𝑆(𝑧) ≥ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑧 → 𝑦), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴. Now, 𝑉𝑆(𝑧 → 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → (𝑦 → 𝑧), 𝑉𝑆(𝑥)}. If 𝑥 → (𝑦 → 𝑧) = 1, then we have 𝑉𝑆 (𝑧 → 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(1), 𝑉𝑆(𝑥)} = 𝑉𝑆 (𝑥). So, 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. Conversely, since 𝑥 → (𝑥 → 1) = 1 for all 𝑥 ∈ 𝐴. Then 𝑉𝑆 (1) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑥)} = 𝑉𝑆(𝑥). On the other hand, from (𝑥 → 𝑦) → (𝑥 → 𝑦) = 1. It follows that 𝑉𝑆 (𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆 (𝑥)}. From the definition 3. 1, 𝑆 is the anti vague filter of 𝐴.∎ From (i) of the proposition 2.2, and the proposition 3.5, we have the following. S. Yahya Mohamed and P. Umamaheswari 220 Corollary 3.6 Let 𝑆 be vague set of BL- algebra 𝐴, 𝑆 be an anti vague filter of 𝐴 if and only if 𝑥 ∗ 𝑦 ≤ 𝑧 or 𝑦 ∗ 𝑥 ≤ 𝑧 implies 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴. Proposition 3.7 Let 𝑆 be a vague set of BL- algebra 𝐴, 𝑆 be an anti vague filter of 𝐴 if and only if (i) 𝑥 ≤ 𝑦, then 𝑉𝑆(𝑥) ≥ 𝑉𝑆 (𝑦), (ii) 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, ∈ 𝐴. Proof: Let 𝑆 be an anti vague filter of BL- algebra 𝐴. Then, from the proposition 3.4, we have 𝑥 ≤ 𝑦, 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦). Since 𝑥 ∗ 𝑦 ≤ 𝑥 ∗ 𝑦 and corollary 3.6, we have 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. Conversely, let 𝑆 be a vague set and satisfies (i) and (ii). For all 𝑥, 𝑦, 𝑧 ∈ 𝐴, if 𝑥 ∗ 𝑦 ≤ 𝑧, then from (i) and (ii), we get 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. From corollary 3.6, we have 𝑆 is an anti vague filter.∎ Proposition 3.8 Let 𝑆 be a vague set of BL- algebra𝐴. Let 𝑆 be an anti vague filter of 𝐴. The following holds for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) If 𝑉𝑆(𝑥 → 𝑦) = 𝑉𝑆 (1), then 𝑉𝑆(𝑥) ≥ 𝑉𝑆 (𝑦), (ii) 𝑉𝑆(𝑥 ∨ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}, (iii) 𝑉𝑆(𝑥 ∗ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}, (iv) 𝑉𝑆(1) = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑥 −)}, (v) 𝑉𝑆(𝑥 → 𝑧) ≤ 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑦 → 𝑧)}, (vi) 𝑉𝑆(𝑥 → 𝑦) ≥ 𝑉𝑆(𝑥 ∗ 𝑧 → 𝑦 ∗ 𝑧), (vii) 𝑉𝑆(𝑥 → 𝑦) ≥ 𝑉𝑆((𝑦 → 𝑧) → (𝑥 → 𝑧)), (viii) 𝑉𝑆(𝑥 → 𝑦) ≥ 𝑉𝑆((𝑧 → 𝑥) → (𝑧 → 𝑦)). Proof: (i) Let 𝑆 be an anti vague filter of BL- algebra 𝐴. Then, from the definition 3.1, and since 𝑉𝑆(𝑥 → 𝑦) = 𝑉𝑆 (1). We have 𝑉𝑆(𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑥 → 𝑦)} Properties of an anti-vague filter in BL- algebras 221 = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(1)} = 𝑉𝑆(𝑥). Thus, 𝑉𝑆 (𝑥) ≥ 𝑉𝑆 (𝑦). (ii) Since 𝑥 ∨ 𝑦 ≥ 𝑥 and 𝑥 ∨ 𝑦 ≥ 𝑦. From the proposition 3.4, we get 𝑉𝑆(𝑥 ∨ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. From the definition 3.1 we have 𝑉𝑆 (𝑥 ∨ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → (𝑥 ∨ 𝑦)), 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆((𝑥 → 𝑥) ∨ (𝑥 → 𝑦)), 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑥)} ≤ 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥{𝑉𝑆(𝑦 → (𝑥 → 𝑦)), 𝑉𝑆(𝑦)}, 𝑉𝑆 (𝑥)} = 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥{𝑉𝑆(1), 𝑉𝑆(𝑦)}, 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑦), 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} Hence, 𝑉𝑆(𝑥 ∨ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. (iii) From (ii) of proposition 3.7, we have 𝑉𝑆 (𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. Since 𝑥 ∗ 𝑦 ≥ 𝑥 ∨ 𝑦, proposition 3.4, and (ii), we have 𝑉𝑆(𝑥 ∗ 𝑦) ≥ 𝑉𝑆(𝑥 ∨ 𝑦) = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. Thus, 𝑉𝑆(𝑥 ∗ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. (iv) From (iii), we have 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑥 −)} = 𝑉𝑆(𝑥 ∗ 𝑥 −) = 𝑉𝑆(1). Therefore, 𝑉𝑆(1) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑥 −)}. (v) From (iii) and proposition 3.4, since (𝑥 → 𝑦) ∗ (𝑦 → 𝑧) ≤ 𝑥 → 𝑧, we get 𝑉𝑆((𝑥 → 𝑦) ∗ (𝑦 → 𝑧)) ≥ 𝑉𝑆((𝑥 → 𝑧), 𝑟𝑚𝑎𝑥{𝑉𝑆((𝑥 → 𝑦), 𝑉𝑆(𝑦 → 𝑧))} ≥ 𝑉𝑆 ((𝑥 → 𝑧). Therefore, we have 𝑉𝑆(𝑥 → 𝑧) ≤ 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑦 → 𝑧)}. From the proposition 2.2 and (i) of proposition 3.7 we can prove (vi), (vii) and (viii) easily.∎ S. Yahya Mohamed and P. Umamaheswari 222 Proposition 3.9 Let 𝑆 be a vague set of BL- algebra A, 𝑆 be an anti vague filter of A if and only if (i) 𝑉𝑆(1) ≤ 𝑉𝑆(𝑥), (ii) 𝑉𝑆 ((𝑥 → (𝑦 → 𝑧)) → 𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, ∈ 𝐴. Proof: Let S be an anti vague filter of A. By the definition 3.1, (i) is straight forward. Since, 𝑉𝑆 ((𝑥 → (𝑦 → 𝑧)) → 𝑧) ≤ 𝑟𝑚𝑎𝑥 {𝑉𝑆 ((𝑥 → (𝑦 → 𝑧)) → (𝑦 → 𝑧)) , 𝑉𝑆 (𝑦)}. (3.1) Now, we have (𝑥 → (𝑦 → 𝑧)) → (𝑦 → 𝑧) = 𝑥 ∨ (y → z) ≥ 𝑥. 𝑉𝑆((𝑥 → (𝑦 → 𝑧)) → (𝑦 → 𝑧)) ≤ VS(𝑥). [from the proposition 3.4] (3.2) Using (3.2) in (3.1), we have 𝑉𝑆((𝑥 → (𝑦 → 𝑧)) → 𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. Conversely, suppose (i) and (ii) hold. Since 𝑉𝑆(𝑦) = 𝑉𝑆(1 → 𝑦) = 𝑉𝑆(((𝑥 → 𝑦) → (𝑥 → 𝑦) → 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑦). From (i), S is an anti vague filter of 𝐴. ∎ Proposition 3.10 Intersection of two anti vague filters of 𝐴 is also an anti vague filter of 𝐴. Proof: Let 𝑈 and 𝑊 be two anti vague filters of 𝐴. To Prove: 𝑈 ∩ 𝑊 is an anti vague filter of 𝐴. For all 𝑥, 𝑦, 𝑧 ∈ 𝐴 such that 𝑧 ≤ 𝑥 → 𝑦, then 𝑧 → (𝑥 → 𝑦) = 1. Since, 𝑈, 𝑊 are two anti vague filters 𝐴, we have 𝑉𝑈 (𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑈 (𝑧), 𝑉𝑈 (𝑥)} and 𝑉𝑊(𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑊(𝑧), 𝑉𝑊(𝑥)}. That is,𝑡𝑈 (𝑦) ≤ max {𝑡𝑈 (𝑧), 𝑡𝑈 (𝑥)} and 1 − 𝑓𝑈 (𝑦) ≤ max{1 − 𝑓𝑈 (𝑧), 1 − 𝑓𝑈 (𝑥)} , 𝑡𝑊(𝑦) ≤ max{𝑡𝑊(𝑧), 𝑡𝑊(𝑥)} and 1 − 𝑓𝑊(𝑦) ≤ max{1 − 𝑓𝑊(𝑧), 1 − 𝑓𝑊(𝑥)}. Since, 𝑡𝑈∩𝑊(𝑦) = min{𝑡𝑈 (𝑦), 𝑡𝑊(𝑦)} Properties of an anti-vague filter in BL- algebras 223 ≤ max {max{𝑡𝑈 (𝑧), 𝑡𝑈 (𝑥)} , max{𝑡𝑊(𝑧), 𝑡𝑊(𝑥)} } = max {max{𝑡𝑈 (𝑧), 𝑡𝑊(𝑧)} , max{𝑡𝑈 (𝑥), 𝑡𝑊(𝑥)} } = max {max{𝑡𝑈∩𝑊(𝑧), 𝑡𝑈∩𝑊(𝑥)} } and 1 − 𝑓𝑈∩𝑊 (𝑦) = max{1 − 𝑓𝑈 (𝑦), 1 − 𝑓𝑊(𝑦)} ≤ max {max{1 − 𝑓𝑈 (𝑧), 1 − 𝑓𝑈 (𝑥)} , max{1 − 𝑓𝑊(𝑧), 1 − 𝑓𝑊 (𝑥)} } = max {max{1 − 𝑓𝑈 (𝑧), 1 − 𝑓𝑊(𝑧)} , max{1 − 𝑓𝑈 (𝑥), 1 − 𝑓𝑊(𝑥)} } = max{max{1 − 𝑓𝑈∩𝑊(𝑧), 1 − 𝑓𝑈∩𝑊(𝑥)} }. Hence, 𝑉𝑈∩𝑊 (𝑦) = [𝑡𝑈∩𝑊(𝑦), 1 − 𝑓𝑈∩𝑊(𝑦)] ≤ 𝑟𝑚𝑎𝑥{𝑉𝑈∩𝑊(𝑧), 𝑉𝑈∩𝑊(𝑥)}. Thus 𝑈 ∩ 𝑊 is an anti vague filter of 𝐴. ∎ Corollary 3.11 Let 𝑅𝑗 be a family of anti vague filters of 𝐴, where 𝑗 ∈ 𝐼, 𝐼 is a index set, then ⋂ 𝑅𝑗𝑗∈𝐼 is an anti vague filter of 𝐴. Note: Union two anti vague filters of BL- algebra 𝐴 need not be an anti vague filter of 𝐴. Proposition 3.12 A 𝜌- vague set and zero vague set of a BL-algebra 𝐴 are anti vague filters of 𝐴. Proof: Let 𝑆 be a 𝜌-vague set of BL-algebra 𝐴, and 𝑆 be an anti vague filter of 𝐴. Then, from the proposition 3.4, we have if 𝑥 ≤ 𝑦, then 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦) for all 𝑥, 𝑦, ∈ 𝐴. To prove: 𝑉𝑆 (𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)} for all 𝑥, 𝑦, ∈ 𝐴. Now,𝑡𝑆 (𝑥 ∗ 𝑦) = 𝜌 = max{𝜌, 𝜌} = max { 𝑡𝑆 (𝑥), 𝑡𝑆 (𝑦)} (3.3) and 1 − 𝑓𝑆 (𝑥 ∗ 𝑦) = 𝜌 = max{𝜌, 𝜌} = max {1 − 𝑓𝑆 (𝑥), 1 − 𝑓𝑆 (𝑦)} for all 𝑥, 𝑦, ∈ 𝐴 (3.4) From (3.3) and (3.4), we have 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. Thus, 𝜌- vague set is an anti vague filter of 𝐴. Similarly, we prove zero vague set is an anti vague of 𝐴. ∎ S. Yahya Mohamed and P. Umamaheswari 224 Theorem 3.13 Let 𝑆 be a vague set of BL-algebra 𝐴, 𝑆 be an anti vague filter of 𝐴 if and only if the set 𝑆(𝜌,𝜎) is either empty or a filter of 𝐴 for all 𝜌, 𝜎 ∈ [0, 1], where 𝜌 ≤ 𝜎. Proof: Let 𝑆 be an anti vague filter of BL-algebra 𝐴 and 𝑆(𝜌,𝜎) ≠ ∅ for all 𝜌, 𝜎 ∈ [0, 1]. To prove: 𝑆(𝜌,𝜎) is a filter of 𝐴. If 𝑥 ≤ 𝑦 and 𝑥 ∈ 𝑆(𝜌,𝜎). From the proposition 3.12, we have 𝑉𝑆(𝑦) ≤ 𝑉𝑆(𝑥) ≤ [𝜌, 𝜎] for all 𝑥, 𝑦 ∈ 𝐴. Thus, 𝑦 ∈ 𝑆(𝜌,𝜎). If 𝑥, 𝑦 ∈ 𝑆(𝜌,𝜎), then 𝑉𝑆 (𝑥) and 𝑉𝑆(𝑦) ≤ [𝜌, 𝜎]. From (ii) of the proposition 3.7, we have 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} ≤ [𝜌, 𝜎]. Thus, 𝑥 ∗ 𝑦 ∈ 𝑆(𝜌,𝜎). Hence 𝑆(𝜌,𝜎) is a filter of 𝐴. Conversely, if for all 𝜌, 𝜎 ∈ [0, 1], the set 𝑆(𝜌,𝜎) is either empty or a filter of 𝐴. Let 𝑡𝑆 (𝑥) = 𝜌1, 𝑡𝑆 (𝑦) = 𝜌2, 1 − 𝑓𝑆 (𝑥) = 𝜎1 and 1 − 𝑓𝑆 (𝑦) = 𝜎2. Put 𝜌 = max{𝜌1, 𝜌2} and 𝜎 = max{1 − 𝜎1, 1 − 𝜎2}. Then, 𝑡𝑆 (𝑥) , 𝑡𝑆 (𝑦) ≤ 𝜌 and 1 − 𝑓𝑆 (𝑥), 1 − 𝑓𝑆 (𝑦) ≤ 𝜎. Thus, 𝑉𝑆(𝑥) and 𝑉𝑆(𝑦) ≤ [𝜌, 𝜎], that is 𝑥, 𝑦 ∈ 𝑆(𝜌,𝜎). Thus, 𝑆(𝜌,𝜎) ≠ ∅. Hence, by the assumption 𝑆(𝜌,𝜎) is a filter of 𝐴. To prove: 𝑆 is an anti vague filter of 𝐴. If 𝑥 ≤ 𝑦, 𝑡𝑆 (𝑥) = 𝜌 and 1 − 𝑓𝑆 (𝑥) = 𝜎. Then 𝑥 ∈ 𝑆(𝜌,𝜎). Since, 𝑆(𝜌,𝜎) is a filter, 𝑦 ∈ 𝑆(𝜌,𝜎), that is, 𝑉𝑆(𝑦) ≤ [𝜌, 𝜎]. (3.5) Since, 𝑆(𝜌,𝜎) is filter of 𝐴, 𝑥 ∗ 𝑦 ∈ 𝑆(𝜌,𝜎). That is, 𝜗𝑆 (𝑥 ∗ 𝑦) ≤ [𝜌, 𝜎] for all 𝑥, 𝑦 ∈ 𝐴 Properties of an anti-vague filter in BL- algebras 225 = [max{𝜌1, 𝜌2}, max{1 − 𝜎1, 1 − 𝜎2}] = 𝑟𝑚𝑎𝑥{[𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)], [𝑡𝑆 (𝑦), 1 − 𝑓𝑆 (𝑦)] = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦 ∈ 𝐴. (3.6) From (3.5) and (3.6), 𝑆 is an anti vague filter of 𝐴. ∎ Note. The filter 𝑆(𝜌,𝜎) is called a vague-cut filter of BL- algebra 𝐴. Proposition 3.14 Let 𝑆 be an anti vague filter of BL-algebra 𝐴. Then 𝑆𝜌 is either empty or a filter of 𝐴 for all 𝜌 ∈ [0, 1]. Proof: Let 𝑆 be an anti vague filter of BL-algebra ℬ. Then from the theorem 3.13, the proof is obvious. ∎ 4. Conclusion In the present paper the notion of an anti-vague filter in BL- algebra with suitable examples are studied. Also investigated some related properties with the help of more implication of an anti-vague filter of BL-algebra. References [1]. R.Biswas, Vague groups, International Journal of Computational Cognition, Vol. 4(2) (2006), 20-23. [2]. C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467- 490. [3]. T. Eswarlal, Vague ideals and normal vague ideals in semirings, International Journal of Computational Cognition, Vol. 6, (2008), 60- 65. [4]. W. L. Gau, D. J. Buehrer, Vague sets, IEEE Transactions on Systems, Man and Cybernetics, Vol. 23 (2), (1993), 610-614. [5]. P. Hảjek, Metamathematics of fuzzy logic, Klower Academic Publishers, Dordrecht, 1999. [6]. L. Z. Liu, K. T. Li, Fuzzy filters of BL-algebras, Information Sciences, 173 (2005), 141-154. S. Yahya Mohamed and P. Umamaheswari 226 [7]. S. Yahya Mohamed and P. Umamaheswari, Vague Filter of BL- Algebras, Journal of Computer and Mathematical Sciences, 9(8), (2018), 914-920. [8]. S. Yahya Mohamed and P. Umamaheswari, Vague prime and Boolean filters of BL- Algebras, Journal of Applied Science and Computations, 5(11),(2018), 470-474. [9]. S. Yahya Mohamed and P. Umamaheswari, Vague implicative filters of BL- algebras, American International Journal of Research in Science, Technology, Engineering and & Mathematics, Conference Proceeding of ICOMAC-2019, 295-299. [10]. S. Yahya Mohamed and P. Umamaheswari, Vague Positive Implicative filter of BL- algebras, Malaya Journal of Matematik, 8(1), (2020), 166-170. [11]. E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic 40 (2001), 467-473. [12]. L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-353. [13]. X. H. Zhang, Fuzzy logic and its algebraic analysis, Science Press, Beijing (2008).