(, q)-interval-valued Fuzzy Dot d-ideals of d-algebras

Transcription

1 Advanced Trends in Mathematics Online: ISSN: X, Vol. 3, pp 1-15 doi: / 015 SciPress Ltd., Switzerland (, q)-interval-valued Fuzzy Dot d-ideals of d-algebras S. R. Barbhuiya 1, K. D. Choudhury 1 Department of Mathematics, Srikishan Sarda College, Hailakandi , Assam, India Department of Mathematics Assam University, Silchar , Assam,India saidurbarbhuiya@gmail.com Keywords: d-algebra, Interval-valued Fuzzy d-ideal, Interval-valued Fuzzy Dot d-ideal, (, )-intervalvalued Fuzzy Dot d-ideal, (, q)-interval-valued Fuzzy Dot d-ideal. Abstract. The concept of (, q)-interval-valued fuzzy dot d-ideals in d-algebras is introduced. Relationship among interval-valued fuzzy d-ideal, interval-valued fuzzy dot d-ideal, (, )-intervalvalued fuzzy d-ideal,(, )-interval-valued fuzzy dot d-ideal, and (, q)-interval-valued fuzzy dot d-ideals are discussed. Conditions for an interval-valued fuzzy d-ideal to be an (, q)-intervalvalued fuzzy dot d-ideals are given. Some properties of interval-valued fuzzy relations and intervalvalued fuzzy ideals under homomorphism are investigated. 1 Introduction In 1991 Xi [1] applied the concept of fuzzy sets to BCK-algebras which are introduced by Imai and Iseki[7]in Neggers and Kim [11]introduced the class of d-algebras which is a generalisation of BCK-algebras and investigated relation between d-algebras and BCK- algebras. Akram and Dar[1] introduced the concepts fuzzy d-algebra, they introduced fuzzy subalgebra and fuzzy d- ideals of d-algebras. Kim [8]introduced the notion of a fuzzy dot subalgebra of d-algebra and investigated some related properties. Bhakat and Das [5, 6] used the relation of belongs to and quasicoincident between fuzzy point and fuzzy set to introduced the concept of(, q)-fuzzy subgroup and (, q)-fuzzy subring. Al-Shehrie[] introduced the notion of fuzzy dot d-ideals of a d-algebra. Interval-valued fuzzy sets were first introduced by Zadeh [15] in After that many researchers consider the interval-valued fuzzification of ideals and subalgebras in BG/ BCK-algebras. The concept of (, q)-interval-valued fuzzification of ideals in ring was introduced in [9]. Here in this paper, we introduce the notion of (, q)-interval-valued fuzzy dot d-ideal of d-algebra and then we investigates some of its interesting properties. Preliminaries Definition 0.1 [1, 8] A d-algebra is a non-empty set X with a constant 0 and a binary operation satisfying the following axioms: (i) x x = 0 (ii) 0 x = 0 (iii) x y = 0 and y x = 0 x = y for all x, y X. For brevity we also call X a d-algebra. Definition 0. [8] A non-empty subset S of a d-algebra X is called a subalgebra of X if x y S, for all x, y S. Definition 0.3 [] A nonempty subset I of a d-algebra X is called an ideal of X if (i) 0 I (ii) x y I and y I x I (iii) x I and y X x y I SciPress applies the CC-BY 4.0 license to works we publish:

2 Volume 3 Definition 0.4 [, 8] A fuzzy subset µ of X is called a fuzzy dot subalgebra of a d-algebra X if µ(x y) µ(x).µ(y) for all x, y X. Definition 0.5 [] A fuzzy subset µ of X is called a fuzzy d-ideal of X if it satisfies the following conditions: (i) µ(0) µ(x) (ii) µ(x) min{µ(x y), µ(y)} (iii) µ(x y) min{µ(x), µ(y)} Definition 0.6 [] A fuzzy subset µ of X is called a fuzzy dot d-ideal of X if it satisfies the following conditions: (i) µ(0) µ(x) (ii) µ(x) µ(x y).µ(y) (iii) µ(x y) µ(x).µ(y) for all x, y X. Definition { 0.7 [5] A fuzzy set µ of the form t if y = x, t ( 0, 1] µ(y) = 0 if y x is called a fuzzy point with support x and value t and it is denoted by x t. Definition 0.8 [5] Let µ be a fuzzy set in X and x t be a fuzzy point then (i) If µ(x) t then we say x t belongs to µ and write x t µ (ii)if µ(x) + t > 1 then we say x t quasi coincidenc µ and write x t qµ (iii)if x t qµ x t µ or x t qµ (iv) If x t qµ x t µ and x t qµ The symbol x t αµ means x t αµ does not hold and q means q For a fuzzy point x t. and a fuzzy set µ in set X,Pu and Liu [10]gave meaning to the symbol x t αµ where α {, q, q, q} Definition 0.9 A fuzzy subset µ of X is called a (, q)-fuzzy d-ideal of X if it satisfies the following conditions: (i)x t µ 0 t qµ (ii)(x y) t, y s µ x m(t,s) qµ (iii)x t, y s µ (x y) m(t,s) qµ t, s (0, 1], x, y X Where m(t, s) = min{t, s} Definition 0.10 A fuzzy subset µ of X is called a (, q)-fuzzy dot d-ideal of X if it satisfies the following conditions: 1. x t µ 0 t qµ. (x y) t, y s µ x t.s qµ 3. x t, y s µ (x y) t.s qµ t, s (0, 1], x X Example 0.11 Consider d-algebra X = {0, a, b, c} with the following cayley table.

3 Advanced Trends in Mathematics Vol. 3 3 * 0 a b c a a 0 0 a b b b 0 0 c c c c 0 Define µ : X [0, 1] by µ by µ(0) = 0.9, µ(a) = µ(b) = 0.8, µ(c) = 0.7, then it is easy to verify that µ is (, q)-fuzzy dot d-ideal X. Definition 0.1 [] Let λ and µ be two fuzzy sets in a set X.The their cartesian product λ µ : X X [0, 1] is defined by (λ µ)(x, y) = λ(x).µ(y), for all x, y X. Let σ be a fuzzy subset of X, then the strongest fuzzy σ relation on d algebra X is the fuzzy subset µ σ of X X given by µ σ (x, y) = σ(x).σ(y) x, y X. A fuzzy relation µ on d algebra X is called a fuzzy σ product relation if µ(x, y) σ(x).σ(y) x, y X. A fuzzy relation µ on d algebra X is called a left fuzzy relation on σ if µ(x, y) = σ(x) x, y X. Note that a left fuzzy relation on σ is a fuzzy σ product relation. Remark 0.13 If X and Y be two d-algebras, then X X is also a d-algebra under the binary operation defined in X X by (x, y) (p, q) = (x p, y q) for all (x, y), (p, q) X X. 3 (, q)-interval-valued fuzzy sets The notion of interval-valued fuzzy set was introduced by L.A.Zadeh[5]. To consider the notion of interval-valued fuzzy sets, we need the following notations.by an interval number â, we mean an interval [a, a], where 0 a a 1. Let D[0, 1] denote the set of all such interval numbers of [0, 1]. Define on D[0, 1] the relations, =, <,. by 1. â 1 â a 1 a and a 1 a. â 1 = â a 1 = a and a 1 = a 3. â 1 < â a 1 < a and a 1 < a 4. â 1.â [min(a 1 a, a 1 a, a 1 a, a 1 a ), max(a 1 a, a 1 a, a 1 a, a 1 a )] = [a 1 a, a 1 a ] 5. kâ = [ka, ka] where 0 k 1 Now consider two intervals â 1 = [a 1, a 1 ], â = [a, a ] D[0, 1] then we define refine minimum rmin as rmin(â 1, â ) = [min(a 1, a ), min(a 1, a )] and refine maximum as rmax rmax(â 1, â ) = [max(a 1, a ), max(a 1, a )] generally if â i = [a 1, a i ], ˆb i = [b 1, b i ] D[0, 1] for i= 1,,3,...then we define rmax(â i, ˆb i ) = [max(a i, b i ), max(a i, b i )] and rmin(â i, ˆb i ) = [min(a i, b i ), min(a i, b i )] and rinf i (â i ) = [ i a i, i a i ] and rsup i (â i ) = [ i a i, i a i ] (D[0, 1], ) is a complete lattice with = rmin, = rmax, ˆ0 = [0 0] and ˆ1 = [1 1] being the least and the greatest element respectively. Definition 0.14 An interval-valued fuzzy set defined on a non empty set X as an objects having the form ˆµ = {x, [µ(x), µ(x)]}, x X where µ and µ are two fuzzy sets in X such that µ(x) µ(x) for all x X. Let ˆµ(x) = [µ(x), µ(x)], x X. Then ˆµ(x) D[0 1], x X If ˆµ and ˆν be two interval-valued fuzzy sets in X, then we define ˆµ ˆν for all µ(x) ν(x) and µ(x) ν(x).

4 4 Volume 3 ˆµ = ˆν for all µ(x) = ν(x) and µ(x) = ν(x). (ˆµ ˆν)(x) = ˆµ(x) ˆν(x) = [max{µ(x), ν(x)}, max{µ(x), ν(x)}]. (ˆµ ˆν)(x) = ˆµ(x) ˆν(x) = [min{µ(x), ν(x)}, min{µ(x), ν(x)}]. ˆµ c (x) = [1 µ(x), 1 µ(x)]. Definition 0.15 Let ˆµ be an interval-valued fuzzy set in X.Then, for every [0, 0] < ˆt [1, 1], the crisp set ˆµ t = {x X ˆµ(x) ˆt} is called the level subset of ˆµ. Definition 0.16 A interval-valued fuzzy subset ˆµ of X is called an interval-valued fuzzy dot subalgebra of a d-algebra X if ˆµ(x y)) ˆµ(x).ˆµ(y) for all x, y X. Definition 0.17 An interval-valued fuzzy set ˆµ in d-algebra X is called an interval-valued fuzzy ideal of X if it satisfies (i) ˆµ(0) ˆµ(x) (ii) ˆµ(x) rmin{ˆµ(x y), ˆµ(y)} for all x, y X (ii) ˆµ(x y) rmin{ˆµ(x), ˆµ(y)} for all x, y X Definition 0.18 An interval-valued fuzzy set ˆµ in d-algebra X is called an interval-valued fuzzy dot d-ideal of X if it satisfies (i) ˆµ(0) ˆµ(x) (ii) ˆµ(x) ˆµ(x y).ˆµ(y) for all x, y X (ii) ˆµ(x y) ˆµ(x).ˆµ(y) for all x, y X Example 0.19 Consider d-algebra X = {0, a, b, c} with the following cayley table. * 0 a b c a a 0 0 a b b b 0 0 c c c c 0 Define ˆµ : X D[0, 1] by ˆµ(0) = [0.8, 0.9], ˆµ(a) = ˆµ(b) = [0.6, 0.8], ˆµ(c) = [0.35, 0.5], then it is easy to verify that ˆµ is (, q)-interval-valued fuzzy dot d-ideal X. Definition 0.0 Let ˆµ(x) = [µ(x), µ(x)] and ˆt = [t, t] D[0, 1], then we define ˆµ(x) + ˆt = [µ(x) + t, µ(x) + t] x X. In particular µ(x) + t > 1, we write ˆµ(x) + ˆt > [1, 1]. Let x X. An interval-valued { fuzzy set ˆµ of the form ˆt if y = x, ˆt D( 0, 1] ˆµ(y) = ˆ0 if y x is called an interval-valued fuzzy point with support x and value ˆt and it is denoted by xˆt. Definition 0.1 ([5]) Let ˆµ be an interval-value fuzzy set in X and xˆt be an interval-value fuzzy point then (i) If ˆµ(x) ˆt then we say xˆt belongs to ˆµ and write xˆt ˆµ (ii)if ˆµ(x) + ˆt > [1, 1] then we say xˆt quasi coincidenc ˆµ and write xˆtqˆµ

5 Advanced Trends in Mathematics Vol. 3 5 (iii)if xˆt qˆµ x t ˆµ or xˆtqˆµ (iv) If xˆt qˆµ xˆt ˆµ and xˆtqˆµ The symbol xˆtαˆµ means xˆtαˆµ does not hold and q means q For an interval-valued fuzzy point xˆt and an interval-valued fuzzy set ˆµ in set X, Pu and Liu [10]gave meaning to the symbol xˆtαˆµ where α {, q, q, q} Definition 0. A interval-valued fuzzy set ˆµ of a d-algebra Xis said to be (α, β)-interval-valued fuzzy d-ideal of X, Where α q if (i) xˆtαˆµ 0ˆtβ ˆµ (ii) (x y)ˆt, yŝαˆµ x rmin(ˆt,ŝ)β ˆµ (iii) (x)ˆt, yŝαˆµ (x y) rmin(ˆt,ŝ)β ˆµ for all x, y X where [0, 0] < ˆt, ŝ [1, 1], where α, β {, q, q, q} Definition 0.3 A interval-valued fuzzy set ˆµ of a d-algebra Xis said to be (α, β)-interval-valued fuzzy dot d-ideal of X, Where α q if (i) xˆtαˆµ 0ˆtβ ˆµ (ii) (x y)ˆt, yŝαˆµ x (ˆt.ŝ)β ˆµ (iii) (x)ˆt, yŝαˆµ (x y) (ˆt.ŝ)β ˆµ for all x, y X where [0, 0] < ˆt, ŝ [1, 1] Definition 0.4 A interval-valued fuzzy set ˆµ of a d-algebra Xis said to be (, q)-intervalvalued fuzzy dot d-ideal of X, if (i) xˆt ˆµ 0ˆt qˆµ (ii) (x y)ˆt, yŝ ˆµ x (ˆt.ŝ) qˆµ (iii) (x)ˆt, yŝ ˆµ (x y) (ˆt.ŝ) qˆµ for all x, y X where [0, 0] < ˆt, ŝ [1, 1] Example 0.5 Consider d-algebra X as in Example 0.11 and ˆµ by ˆµ(0) = [0.75, 0.9], ˆµ(a) = ˆµ(b) = [0.7, 0.8], ˆµ(c) = [0.65, 0.7], then it is easy to verify that ˆµ is (, q)-interval-valued fuzzy dot d-ideal X. Theorem 0.6 Every (, )-interval-valued fuzzy d-ideal of d algebra X is an (, )-intervalvalued fuzzy dot d-ideal of X. Proof Straightforward Remark 0.7 The converse of Theorem 0.6 is not true as shown in following Example. Example 0.8 Consider d-algebra X = {0, a, b} with the following cayley table. * 0 a b a b 0 b b a a 0

6 6 Volume 3 Define ˆµ : X D[0, 1] by ˆµ(0) = [0.6, 0.7], ˆµ(a) = [0.7, 0.8], ˆµ(b) = [0.8, 0.9] then it is easy to verify that ˆµ is (, )-interval-valued fuzzy dot d ideal X. Here (a b) [0.77,0.88], b [0.77,0.88] ˆµ But a [0.77,0.88] ˆµ. Therefore ˆµ is not an (, )-interval-valued fuzzy d-ideal X. Theorem 0.9 Every interval-valued fuzzy d-ideal of a d algebra X is an interval-valued fuzzy dot d-ideal of X. Remark 0.30 The converse of Theorem 0.9 is not true as shown in following Example. Example 0.31 Consider d-algebra X and ˆµ as in Example 0.8 then it is easy to verify that ˆµ is an interval-valued fuzzy dot d ideal X. But rmin{ˆµ(a b), ˆµ(b)} = ˆµ(b) = [0.8, 0.9] ˆµ(a) = [0.7, 0.8] Therefore ˆµ is not an interval-valued fuzzy d-ideal X. Theorem 0.3 An interval-valued fuzzy subset ˆµ of a d algebra X is a interval-valued fuzzy d-ideal iff ˆµ is an (, )-interval-valued fuzzy d-ideal of X Proof Let ˆµ be an interval-valued fuzzy d-ideal of X. To prove ˆµ is an (, )-interval-valued fuzzy d-ideal of X. Let x X such that xˆt ˆµ where ˆt D(0, 1) then ˆµ(x) ˆt ˆµ(0) ˆµ(x) ˆt [ Since ˆµ-is interval-valued fuzzy d-ideal ] ˆµ(0) ˆt 0ˆt ˆµ xˆt ˆµ 0ˆt ˆµ Again let x, y X such that (x y)ˆt, yŝ ˆµ where ˆt, ŝ D(0, 1) then ˆµ(x y) ˆt, ˆµ(y) ŝ Now ˆµ(x) rmin{ˆµ(x y), ˆµ(y)} rmin(ˆt, ŝ)[ Since ˆµ-is interval-valued fuzzy d-ideal ] x rmin(ˆt,ŝ) ˆµ (x y)ˆt, yŝ ˆµ x rmin(ˆt,ŝ) ˆµ Again let xˆt, yŝ ˆµ ˆµ(x) ˆt, ˆµ(y) ŝ Now ˆµ(x y) rmin{ˆµ(x), ˆµ(y)} rmin(ˆt, ŝ)[ Since ˆµ-interval-valued fuzzy d-ideal ] (x y) rmin(ˆt,ŝ) ˆµ xˆt, yŝ ˆµ (x y) rmin(ˆt,ŝ) ˆµ Hence ˆµ is an (, )-interval-valued fuzzy d-ideal of X. Conversely suppose ˆµ is an (, )-interval-valued fuzzy d-ideal of X. to prove ˆµ is an interval-valued fuzzy d-ideal of X. Let x X Now since xˆµ(x) ˆµ therefore 0ˆµ(x) ˆµ ˆµ(0) ˆµ(x) [ Since ˆµ is an(, )-interval-valued fuzzy d-ideal of X ] Let x, y X again since (x y)ˆµ(x y) ˆµ and yˆµ(y) ˆµ x rmin{ˆµ(x y),ˆµ(y)} µ[ Since ˆµ is an(, )-interval-valued fuzzy d-ideal of X ] ˆµ(x) rmin{ˆµ(x y), ˆµ(y)} Let x, y X again since (x)ˆµ(x) ˆµ and yˆµ(y) µ (x y) rmin{ˆµ(x),ˆµ(y)} ˆµ[ Since ˆµ is an(, )-interval-valued fuzzy d-ideal of X ] ˆµ(x y) rmin{ˆµ(x), ˆµ(y)} Hence ˆµ is an interval-valued fuzzy d-ideal of X. Theorem 0.33 An interval-valued fuzzy subset ˆµ of a d algebra X is a interval-valued fuzzy dot d-ideal iff ˆµ is a (, )-interval-valued fuzzy dot d-ideal of X

7 Advanced Trends in Mathematics Vol. 3 7 Proof ˆµ be an interval-valued fuzzy dot d-ideal of X.To prove ˆµ is an (, )-interval-valued fuzzy dot d-ideal of X. Let x X such that xˆt ˆµ where ˆt D(0, 1) then ˆµ(x) ˆt ˆµ(0) ˆµ(x) ˆt[ Since ˆµ is an interval-valued fuzzy dot d-ideal ] ˆµ(0) ˆt 0ˆt ˆµ xˆt ˆµ 0ˆt ˆµ Let x, y X such that (x y)ˆt, yŝ ˆµ where ˆt, ŝ D(0, 1) then ˆµ(x y) ˆt, µ(y) ŝ Now ˆµ(x) ˆµ(x y).ˆµ(y) ˆt.ŝ [ Since ˆµ is an interval-valued fuzzy dot d-ideal ] xˆt.ŝ ˆµ (x y)ˆt, yˆt ˆµ xˆt.ŝ ˆµ Again let xˆt, yŝ ˆµ ˆµ(x) ˆt, ˆµ(y) ŝ Now ˆµ(x y) ˆµ(x).ˆµ(y) ˆt.ŝ [ Since ˆµ is an interval-valued fuzzy dot d-ideal ] (x y)ˆt.ŝ ˆµ xˆt, yŝ ˆµ (x y)ˆt.ŝ ˆµ Hence ˆµ is an (, )-interval-valued fuzzy dot d-ideal of X. Conversely suppose ˆµ is an (, )-interval-valued fuzzy dot d-ideal of X. to prove ˆµ is an interval-valued fuzzy dot d-ideal of X. Let x X and ˆt = ˆµ(x) then ˆµ(x) ˆt xˆt ˆµ 0ˆt ˆµ[ Since ˆµ is (, )-is an interval-valued fuzzy dot d-ideal of X.] ˆµ(0) ˆt ˆµ(0) ˆµ(x) Let x, y X and ˆt = ˆµ(x y), ŝ = ˆµ(y) then ˆµ(x y) ˆt, ˆµ(y) ŝ (x y)ˆt ˆµ, yŝ ˆµ xˆt.ŝ ˆµ [ Since ˆµ is (, )-is an interval-valued fuzzy dot d-ideal of X ] ˆµ(x) ˆt.ŝ = ˆµ(x y).ˆµ(y) Again let ˆt = ˆµ(x), ŝ = ˆµ(y) then ˆµ(x) ˆt, ˆµ(y) ŝ xˆt ˆµ, yŝ ˆµ (x y)ˆt.ŝ ˆµ [ Since ˆµ is (, )-interval-valued fuzzy dot d-ideal of X ] ˆµ(x y) ˆt.ŝ = ˆµ(x).ˆµ(y) Hence ˆµ is an interval-valued fuzzy dot d-ideal of X. Theorem 0.34 Every interval-valued fuzzy d-ideal of a d algebra X is an (, )-interval-valued fuzzy dot d-ideal of X. Proof It follows from Theorem 0.9 and Theorem Remark 0.35 The converse of Theorem 0.34 is not true as shown in following Example. Example 0.36 Consider d-algebra X and ˆµ as in Example 0.8 then it is easy to verify that ˆµ is an (, )-interval-valued fuzzy dot d ideal X. But min{ˆµ(a b), ˆµ(b)} = ˆµ(b) = [0.8, 0.9] ˆµ(a) = [0.7, 0.8] Therefore ˆµ is not an interval-valued fuzzy d-ideal X. Theorem 0.37 Every (, )-interval-valued fuzzy d-ideal of a d algebra X is an interval-valued fuzzy dot d-ideal of X. Proof It follows from Theorem 0.3 and Theorem 0.9.

8 8 Volume 3 Remark 0.38 The converse of Theorem0.37 is not true as shown in following Example. Example 0.39 Consider d-algebra X and ˆµ as in Example 0.8 then it is easy to verify that ˆµ is an interval-valued fuzzy dot d ideal X. But (a b) [0.77,0.88], b [0.77,0.88] ˆµ But a [0.77,0.88] ˆµ. Therefore ˆµ is not an (, )-interval-valued fuzzy d-ideal X. Definition 0.40 A fuzzy subset ˆµ of X is called a (, )-interval-valued fuzzy dot subalgebra if xˆt, yˆt ˆµ (x y)ˆt.ŝ ˆµ for all x, y X. Proposition 0.41 Every (, )-interval-valued fuzzy dot d-ideal of a d-algebra X is an (, )- interval-valued fuzzy dot sub algebra of X. Remark 0.4 The converse of Proposition 0.41 is not true as shown in following example. Example 0.43 Consider d algebra X = {0, a, b, c} with the following cayley table. * 0 a b c a a 0 0 a b b b 0 0 c c c a 0 Define ˆµ : X [0, 1] by ˆµ(0) = ˆµ(b) = [0.8, 0.9], ˆµ(a) = ˆµ(c) = [0.7, 0.8] then it is easy to verify that ˆµ is an (, )-interval-valued fuzzy dot subalgebra of X. But ˆµ is not an (, )-interval-valued fuzzy dot d-ideal of X because ˆµ(a b) = ˆµ(0) = [0.8, 0.9] and ˆµ(b) = [0.8, 0.9] therefore ˆµ(a b) [0.8, 0.9] and ˆµ(b) [0.8, 0.9] ˆµ(a) [0.8, 0.9].[0.9, 0.9] = [0.7, 0.81] Theorem 0.44 Every(, )-interval-valued fuzzy dot d-ideal of a d-algebra X is a (, q)-interval-valued fuzzy dot d-ideal X.But the converse is not true as shown in the following Example. Example 0.45 Consider d-algebra X = {0, a, b, c} with the following cayley table. * 0 a b c a a 0 0 a b b b 0 0 c c c a 0 Define ˆµ : X [0, 1] by ˆµ(0) = [0.85, 0.95], ˆµ(a) = ˆµ(b) = [0.8, 0.9], ˆµ(c) = [0.6, 0.7], then it is easy to verify that ˆµ is (, q)-interval-valued fuzzy dot d-ideal X.Since a [0.8,0.9] = (c b) [0.8,0.9], b [0.8,0.9] ˆµ but ˆµ(c) [0.8, 0.9].[0.8, 0.9] i.e c [0.64,0.81] ˆµ Theorem 0.46 If ˆµ, ˆν are (, q)-interval-valued fuzzy dot d-ideal of a d-algebra X, then so is ˆµ ˆν.

9 Advanced Trends in Mathematics Vol. 3 9 Proof Let x, y X such that xˆt (ˆµ ˆν) (ˆµ ˆν)(x) ˆt rmin{ˆµ(x), ˆν(x)} ˆt ˆµ(x) ˆt, ˆν(x) ˆt xˆt ˆµ and xˆt ˆν 0ˆt qˆµ and 0ˆt qˆν [ Since ˆµ, ˆν both are(, q)-intervalvalued fuzzy dot d-ideals of X] 0ˆt q(ˆµ ˆν) Again let (x y)ˆt, yŝ (ˆµ ˆν) (ˆµ ˆν)(x y) ˆt, (ˆµ ˆν)(y) ŝ rmin{ˆµ(x y), ˆν(x y)} ˆt and rmin{ˆµ(y), ˆν(y)} ŝ ˆµ(x y) ˆt, ˆν(x y) ˆt and ˆµ(y) s, ˆν(y) ŝ ˆµ(x y) ˆt, ˆµ(y) s and ˆν(x y) ˆt, ˆν(y) ŝ (x y)ˆt, yŝ ˆµ and (x y)ˆt, y s ˆν xˆt.ŝ qˆµ and xˆt.ŝ qˆν [ Since ˆµ, ˆν both are(, q)-interval-valued fuzzy dot d-ideals of X] xˆt.ŝ q(ˆµ ˆν) Similarly we can prove that xˆt, yŝ (ˆµ ˆν) (x y)ˆt.ŝ q(ˆµ ˆν) Hence the proof. Theorem 0.47 If ˆµ is a (q,q)-interval-valued fuzzy dot d-ideal of a d algebra X then it is also an(, )-interval-valued fuzzy dot d-ideal of X. Proof Let ˆµ be an (q, q)-interval-valued fuzzy dot d-ideal of a d-algebra X, to prove ˆµ is an (, )- interval-valued fuzzy dot d-ideal of X. let x X such thatxˆt ˆµ ˆµ(x) ˆt ˆµ(x) + ˆδ > ˆt where ˆδ is a arbitrary small positive interval in D[0 1] ˆµ(x) + ˆδ ˆt + ˆ1 > ˆ1 (x)ˆ1+ˆδ ˆt qˆµ (0)ˆ1+ˆδ ˆt qˆµ ˆµ(0) + ˆδ ˆt + ˆ1 > ˆ1 ˆµ(0) + ˆδ > ˆt ˆµ(0) ˆt 0ˆt ˆµ Thereforexˆt ˆµ 0ˆt ˆµ Again let x, y X such that (x y)ˆt, yŝ ˆµ therefore ˆµ(x y) t, ˆµ(y) ŝ ˆµ(x y) + ˆδ > ˆt, ˆµ(y) + ˆδ > ŝ where ˆδ is a arbitrary small positive interval in D[0 1] ˆµ(x y) + ˆδ ˆt + ˆ1 > ˆ1, ˆµ(y) + ˆδ ŝ + ˆ1 > ˆ1 (x y)ˆ1+ˆδ ˆt qˆµ and (y)ˆ1+ˆδ ŝ qˆµ (x) (ˆ1+ˆδ ˆt).(ˆ1+ˆδ ŝ) qµ [since µ is (q,q)-interval-valued fuzzy dot d-ideal of X ] ˆµ(x) + (ˆ1 + ˆδ ˆt).(ˆ1 + ˆδ ŝ) > ˆ1 ˆµ(x) + (ˆ1 ˆt).(ˆ1 ŝ) ˆ1 taking ˆδ = ˆ0 ˆµ(x) + ˆ1 ˆt ŝ + ˆt.ŝ ˆ1 ˆµ(x) ŝ + ˆt ˆt.ŝ ˆµ(x) ˆt.ŝ + ˆt.ŝ ˆt.ŝ [ Since ˆt ˆt.ŝ and ŝ ˆt.ŝ] ˆµ(x) ˆt.ŝ xˆt.ŝ ˆµ Again let xˆt, yŝ ˆµ therefore ˆµ(x) ˆt, ˆµ(y) ŝ ˆµ(x) + ˆδ > ˆt, ˆµ(y) + ˆδ > ŝ where ˆδ is a arbitrary small positive interval in D[0 1] ˆµ(x) + ˆδ ˆt + ˆ1 > ˆ1, ˆµ(y) + ˆδ ŝ + ˆ1 > ˆ1 (x)ˆ1+ˆδ ˆt qˆµ and (y)ˆ1+ˆδ ŝ qˆµ (x y) (ˆ1+ˆδ ˆt).(ˆ1+δ s) qˆµ [since ˆµ is (q,q)-interval-valued fuzzy dot d-ideal of X ] ˆµ(x y) + (ˆ1 + ˆδ ˆt).(ˆ1 + ˆδ ŝ) > ˆ1 ˆµ(x y) + (ˆ1 ˆt).(ˆ1 ŝ) ˆ1 taking ˆδ = ˆ0 ˆµ(x y) + ˆ1 ˆt ŝ + ˆt.ŝ ˆ1 ˆµ(x y) ŝ + ˆt ˆt.ŝ ˆµ(x y) ˆt.ŝ + ˆt.ŝ ˆt.ŝ [ Since ˆt ˆt.ŝ and ŝ ˆt.ŝ]

10 10 Volume 3 ˆµ(x y) ˆt.ŝ (x y)ˆt.ŝ ˆµ Hence ˆµ is an (, )-interval-valued fuzzy dot d-ideal of X. Remark 0.48 The converse of Theorem 0.47 is not true as shown in following Example. Example 0.49 Consider d-algebra X and ˆµ as in Example 0.11,then ˆµ is an (, )-interval-valued fuzzy dot d-ideal of X.But ˆµ is not an (q, q)-interval-valued fuzzy dot d-ideal of X because if x = c, y = a, x y = c a = c, t = 0.7, s = 0.4 then µ(x y) + t = > 1, µ(y) + s = > 1 but µ(x) + t.s = = < 1 i.e xˆt.ŝqµ. Theorem 0.50 A fuzzy subset ˆµ of a d-algebra X is an (, q)-interval-valued fuzzy dot d-ideal of X iff (i) ˆµ(0) rmin{ˆµ(x), [ ]} (ii) ˆµ(x) rmin{ˆµ(x y).ˆµ(y), ˆ1 ˆµ(x y).ˆµ(y)} (iii ˆµ(x y) rmin{ˆµ(x).ˆµ(y), ˆ1 ˆµ(x).ˆµ(y)} Proof Let ˆµ be an (, q)-interval-valued fuzzy dot d-ideal of X (i) Assume ˆµ(0) < rmin{ˆµ(x), [0.5, 0.5]} SubCase I: If ˆµ(x) < [0.5, 0.5], them rmin{ˆµ(x), [0.5, 0.5]} = ˆµ(x) ˆµ(0) < ˆµ(x) ˆµ(0) < ˆt < ˆµ(x) for some (0 0) < ˆt < (0.5, 0.5) xˆt ˆµ and 0ˆt ˆµ also ˆµ(0) + ˆt < ˆ1 i.e 0ˆtqµ 0ˆt qˆµ Which is a contradiction SubCase II: If ˆµ(x) [0.5, 0.5] i.e x [0.5, 0.5] ˆµ, them rmin{ˆµ(x), [0.5, 0.5]} = [0.5, 0.5] ˆµ(0) < rmin{ˆµ(x), [0.5, 0.5]} = [0.5, 0.5] 0 [0.5, 0.5] ˆµ and ˆµ(0) + [0.5, 0.5] < [0.5, 0.5] + [0.5, 0.5] = ˆ1i.e0 [0.5, 0.5] qˆµ 0 [0.5, 0.5] qˆµ Which is again a contradiction, therefore Therefore we must have ˆµ(0) rmin{ˆµ(x), [0.5, 0.5]} (ii) Assume ˆµ(x) < rmin{ˆµ(x y).ˆµ(y), ˆ1 ˆµ(x y).µ(y)} choose [0, 0] ˆt, ŝ [0, 1] such that ˆµ(x) < ˆt.ŝ < rmin{ˆµ(x y).ˆµ(y), ˆ1 ˆµ(x y).ˆµ(y)} where ˆµ(x y) ˆt and ˆµ(y) ŝ SubCase I: If ˆµ(x y).ˆµ(y) < [0.5, 0.5], them m{ˆµ(x y).ˆµ(y), ˆ1 ˆµ(x y).ˆµ(y)} = ˆµ(x y).ˆµ(y) ˆµ(x) < ˆt.ŝ < ˆµ(x y).ˆµ(y) < [0.5, 0.5] ˆµ(x y) ˆt and ˆµ(y) ŝ (x y)ˆt ˆµ and yŝ ˆµ butxˆt.ŝ ˆµ and ˆµ(x) + ˆt.ŝ < ˆ1 and xˆt.ŝqˆµ Which contradict the fact that ˆµ is an (, q)-interval-valued fuzzy dot d-ideal of X SubCase II: If ˆµ(x y).ˆµ(y) [0.5, 0.5], them ˆµ(x) < ˆt.ŝ < [0.5, 0.5] < ˆµ(x y).ˆµ(y) ˆµ(x y) ˆt and ˆµ(y) ŝ (x y)ˆt ˆµ and yŝ ˆµ butˆµ(x) ˆt.ŝ and ˆµ(x)+ˆt.ŝ < ˆ1 that is xˆt.ŝ ˆµ and xˆt.ŝqˆµ that is xˆt.ŝ qˆµ Which is again a contradiction, therefore in both subcases Therefore we must haveˆµ(x y) rmin{ˆµ(x).ˆµ(y), ˆ1 ˆµ(x).ˆµ(y)} (iii) Proof is similar to case (ii) above Theorem 0.51 A fuzzy subset ˆµ of d- algebra X is an (, q)-interval-valued fuzzy dot d-ideal of X and ˆµ(x) < [ 5 1] x X then ˆµ is also an (, )-interval-valued fuzzy dot d-ideal of X. Proof Let ˆµ be an (, q)-interval-valued fuzzy dot d-ideal of X and ˆµ(x) < [ 5 1 ] x X Let xˆt ˆµ ˆµ(x) ˆt

11 Advanced Trends in Mathematics Vol ˆt ˆµ(x) < [ 5 1] and also ˆµ(0) < [ 5 1] ˆµ(0) + ˆt < [ 5 1] + [ 5 1 ] = [ 5 1, 5 1] ˆµ(0) + ˆt ˆ1 0ˆtqˆµ therefore xˆt ˆµ 0ˆtqˆµ Since ˆµ be an (, q)-interval-valued fuzzy dot d-ideal of X, therefore we must have xˆt ˆµ 0ˆt ˆµ Again let (x y)ˆt, yŝ ˆµ ˆt ˆµ(x y) < [ 5 1, 5 1 ] and ˆt ˆµ(y) < [ 5 1, ˆt.ŝ ˆµ(x y).ˆµ(y) < [ 5 1 ].[ 5 1 ] and 5 1 ] ˆt.ŝ [ 5 1 ].[ 5 1 ] and ˆµ is also an (, )-interval-valued fuzzy dot d-ideal of X ˆµ(x) < [ 5 1 ] µ(x) + ˆt.ŝ < [ 5 1 ].[ 5 1 ] + [ 5 1 ] = ˆ1 ˆµ(x) + ˆt.ŝ < ˆ1 xˆt.ŝqˆµ therefore (x y)ˆt, yŝ ˆµ xˆt.ŝqˆµ since ˆµ is an (, q)-interval-valued fuzzy dot d-ideal of X,therefore we must have xˆt.ŝ ˆµ. Therefore (x y)ˆt, yŝ ˆµ xˆt.ŝ ˆµ Similarly we can prove that xˆt, yŝ ˆµ (x y)ˆt.ŝ ˆµ Hence ˆµ is an (, )-interval-valued fuzzy dot d-ideal of X. Theorem 0.5 Let ˆµ be a left fuzzy relation on a fuzzy subset ˆσ of a d-algebra X.If ˆµ is an (, )- interval-valued fuzzy dot d-ideal of X X then ˆσ is an (, )-interval-valued fuzzy dot d-ideal of d-algebra X. Proof Suppose that a left fuzzy relation ˆµ on ˆσ is an (, )-interval-valued fuzzy dot d-ideal of X X. To prove that ˆσ is an (, )-interval-valued fuzzy dot d-ideal of X Let x, y X since ˆσ(x) = ˆµ(x, y) let xˆt ˆσ ˆσ(x) ˆt ˆσ(x) = ˆµ(x, y) ˆt (x, y)ˆt ˆµ (0, 0)ˆt ˆµ [ Since ˆµ is an (, )-interval-valued fuzzy dot d-ideal of X X ] ˆµ(0, 0) ˆt ˆσ(0) ˆt 0ˆt ˆσ therefore xˆt ˆσ 0ˆt ˆσ Let x, x, y, y X ˆσ(x) = ˆµ(x, y) now Let (x x )ˆt, (x )ŝ ˆσ ˆσ(x x ) ˆt and ˆσ(x ) ŝ Now ˆσ(x) = ˆµ(x, y) ˆµ((x, y) (x, y )).ˆµ(x, y )[ Since ˆµ is also an interval-valued fuzzy dot d- ideal of X X, see Theorem 0.33 ] = ˆµ(x x, y y ).ˆµ(x, y ) = ˆσ(x x ).ˆσ(x ) ˆt.ŝ (x)ˆt.ŝ ˆσ (x x )ˆt, (x )ŝ ˆσ (x)ˆt.ŝ ˆσ Again let xˆt, x ŝ ˆσ ˆσ(x) ˆt and ˆσ(x ) ŝ ˆσ(x x ) = ˆµ(x x, y y ) = ˆµ((x, y) (x, y )) ˆµ(x, y).ˆµ(x, y ) ˆσ(x).ˆσ(x ) ˆt.ŝ (x x )ˆt.ŝ ˆσ xˆt, x ŝ ˆσ (x x )ˆt.ŝ ˆσ Hence from above ˆσ is an (, )-interval-valued fuzzy dot d-ideal of X

12 1 Volume 3 4 Homomorphism of d-algebras and (, q)-interval-valued Fuzzy Dot d-ideals Definition 0.53 Let X and X be two d-algebras, then a mapping f : X X is said to be homomorphism if f(x y) = f(x) f(y) x, y X. Theorem 0.54 Let X and X be two d-algebras and f : X X be a homomorphism. Then f(0) = 0 Proof Let x X therefore f(x) X. Now f(0) = f(x x) = f(x) f(x) = 0 0 = 0. Theorem 0.55 Let f : X X be an onto homomorphism of d-algebras, ˆν be a (, q)- interval-valued fuzzy dot d-ideal of X, then the pre-image f 1 (ˆν) of ˆν under f is an (, q)-intervalvalued fuzzy dot d-ideal of X. Proof f 1 (ˆν) is defined as f 1 (ˆν)(x) = ˆν(f(x)) x X Let ˆν be an (, )-interval-valued fuzzy dot d-ideal of X Let x X such that xˆt f 1 (ˆν) f 1 (ˆν)(x) ˆt ˆνf(x) ˆt (f(x))ˆt ˆν x ˆν where ˆt f(x) = x 0 ˆt qˆν [Since ˆν is an (, q)-interval-valued fuzzy dot d-ideal of X ] [f(0)]ˆt qˆν ˆνf(0) ˆt or ˆνf(0) + ˆt ˆ1 f 1 (ˆν)(0) ˆt or f 1 (ˆν)(0) + ˆt ˆ1 0ˆt f 1 (ˆν) or 0ˆtqf 1 (ˆν) 0ˆt qf 1 (ˆν) Thereforexˆt f 1 (ˆν) 0ˆt qf 1 (ˆν) Let x, y X such that (x y)ˆt, yŝ f 1 (ˆν) f 1 (ˆν)(x y) ˆt and f 1 (ˆν)(y) ŝ ˆνf(x y) ˆt and ˆνf(y) ˆt (f(x y))ˆt ˆν and (f(y))ŝ ˆν (f(x) f(y))ˆt ˆν and (f(y))ŝ ˆν [ Since f is homomorphism ] (f(x))ˆt.ŝ qˆν [Since ˆν is an (, q)-interval-valued fuzzy dot d-ideal of X ] ˆνf(x) ˆt.ŝ or ˆνf(x) + ˆt.ŝ ˆ1 f 1 (ˆν)(x) ˆt.ŝ or f 1 (ν)(x) + ˆt.ŝ ˆ1 xˆt.ŝ f 1 (ˆν) or xˆt.ŝqf 1 (ˆν) xˆt.ŝ qf 1 (ˆν) Therefore (x y)ˆt, yŝ f 1 (ˆν) xˆt.ŝ qf 1 (ˆν) Again let x, y X such that xˆt, yŝ f 1 (ˆν) f 1 (ˆν)(x) ˆt and f 1 (ˆν)(y) ŝ ˆνf(x) ˆt and ˆνf(y) ŝ (f(x)ˆt ˆν and (f(y))ŝ ˆν (f(x) f(y))ˆt.ŝ qˆν [Since ˆν is an (, q)-interval-valued fuzzy dot d-ideal of X ] ˆν(f(x) f(y)) ˆt.ŝ or ˆν(f(x) f(y)) + ˆt.ŝ ˆ1 ˆν(f(x y)) ˆt.ŝ or ˆν(f(x y)) + ˆt ˆ1 [ Since f is homomorphism ] f 1 (ˆν)(x y) ˆt.ŝ or f 1 (ˆν)(x y) + ˆt.ŝ ˆ1 (x y)ˆt.ŝ f 1 (ˆν) or (x y)ˆt.ŝqf 1 (ˆν) (x y)ˆt.ŝ qf 1 (ˆν) Therefore xˆt, yŝ f 1 (ˆν) (x y)ˆt.ŝ qf 1 (ν) Hence f 1 (ˆν) is an (, q)-interval-valued fuzzy dot d-ideal of X. Theorem 0.56 An onto homomorphic image of an (, q)-interval-valued fuzzy dot d-ideal with the sup property is an (, q)-interval-valued fuzzy dot d-ideal.

13 Advanced Trends in Mathematics Vol Proof Let f : X X be an onto homomorphism of d-algebras, ˆµ be an (, q)-interval-valued fuzzy dot d-ideal of X, and the image of ˆµ under f be f(ˆµ). To prove f(ˆµ) is an (, q)-fuzzy dot d-ideal of X Since 0 f 1 (0 ) therefore, for any x, y X, let x 0, y 0 X such that and then, let ˆµ(x 0 ) = sup ˆµ(z) ˆµ(y) = sup ˆµ(z) z f 1 (x ) z f 1 (y ) ˆµ(x 0 y 0 ) = x ˆt f(ˆµ) f(ˆµ)(x ) ˆt sup ˆµ(z) z f 1 (x y ) sup ˆµ(z) = ˆt z f 1 (x ) ˆµ(x 0 ) ˆt (x 0 )ˆt ˆµ 0ˆt qˆµ [ Since ˆµ be an (, q)-interval-valued fuzzy dot d-ideal of X] ˆµ(0) ˆt or ˆµ(0) + ˆt ˆ1 sup ˆµ(z) ˆt or z f 1 (0 ) sup ˆµ(z) + ˆt ˆ1 z f 1 (0 ) f(ˆµ)(0 ) ˆt or f(ˆµ)(0 ) + ˆt ˆ1 0 f(ˆµ) or qf(ˆµ) ˆt 0 ˆt 0 ˆt qf(ˆµ) Therefore x ˆt f(ˆµ) 0 ˆt qf(ˆµ) Again, let(x y )ˆt, y ŝ f(ˆµ) f(ˆµ)(x y ) ˆt, f(ˆµ)(y ) ŝ sup z f 1 (x y ) ˆµ(z) ˆt, sup z f 1 (y ) ˆµ(x 0 y 0 ) ˆt, ˆµ(y 0 ) ŝ ˆµ(z) ŝ (x 0 y 0 )ˆt ˆµ, (y 0 )ŝ ˆµ (x 0 )ˆt.ŝ qˆµ [ Since ˆµ be an (, q)-interval-valued fuzzy dot d-ideal of X,] ˆµ(x 0 ) ˆt.ŝ or ˆµ(x 0 ) + ˆt.ŝ ˆ1 sup ˆµ(z) ˆt.ŝ or z f 1 (x ) sup ˆµ(z) + ˆt.ŝ ˆ1 z f 1 (x ) f(ˆµ)(x ) ˆt.ŝ or f(ˆµ)(x ) + ˆt.ŝ ˆ1 (x ) ˆ ˆt.ŝ f(ˆµ) or (x )ˆt.ŝqf(µ) (x )ˆt.ŝ qf(ˆµ) (x y )ˆt, y ŝ f(ˆµ) (x )ˆt.ŝ qf(ˆµ) Similarly we can prove x ˆt, y ŝ f(ˆµ) (x y )ˆt.ŝ qf(ˆµ) Hence from above f(ˆµ) is an (, q)-interval-valued fuzzy dot d-ideal of X. Theorem 0.57 Let f : X X be an onto homomorphism of d-algebras, ˆµ be a fuzzy subset of X such that f 1 (ˆµ) is an (, q)- fuzzy dot d-ideal of X, then ˆµ is also an (, q)-interval-valued fuzzy dot d-ideal of X.

14 14 Volume 3 Proof Let f 1 (ˆµ) be an (, q)-interval-valued fuzzy dot d-ideal of X Let x, y X since f is onto so there exists x, y X such that f(x) = x, f(y) = y and also f is homomorphism therefore f(x y) = f(x) f(y) = x y Let x ˆt ˆµ where ˆt D[0 1] Therefore ˆµ(x ) ˆt ˆµ(f(x)) ˆt f 1 (ˆµ)(x) ˆt xˆt f 1 (ˆµ) 0 t qf 1 (ˆµ) [Since f 1 (ˆµ) is an (, q)-interval-valued fuzzy dot d-ideal of X.) f 1 (ˆµ)(0) ˆt or f 1 (ˆµ)(0) + ˆt ˆ1 ˆµ(f(0)) ˆt or ˆµ(f(0)) + ˆt ˆ1 ˆµ(0 ) ˆt or ˆµ(0 ) + ˆt ˆ1 0 ˆµ or qˆµ ˆt 0 ˆt 0 qˆµ ˆt Therefore x ˆµ qˆµ ˆt 0 ˆt Again let (x y )ˆt, y ŝ ˆµ where ˆt, ŝ D[0 1] Therefore ˆµ(x y ) ˆt and ˆµ(y ) ŝ ˆµ(f(x y)) ˆt and ˆµ(f(y)) ŝ f 1 (ˆµ)(x y) ˆt and f 1 (ˆµ)(y) ŝ (x y)ˆt f 1 (ˆµ) and yˆt f 1 (ˆµ) xˆt.ŝ qf 1 (ˆµ) [Since f 1 (ˆµ) is an (, q)-interval-valued fuzzy dot d-ideal of X.) f 1 (ˆµ)(x) ˆt.ŝ or f 1 (ˆµ)(x) + ˆt.ŝ ˆ1 ˆµ(f(x)) ˆt.ŝ or ˆµ(f(x)) + ˆt.ŝ ˆ1 ˆµ(x ) ˆt.ŝ or ˆµ(x ) + ˆt.ŝ ˆ1 x ˆµ or qˆµ ˆt.ŝ x ˆt.ŝ x qˆµ ˆt.ŝ (x y )ˆt, y ŝ ˆµ qˆµ x ˆt.ŝ Again let x, y X such that (x )ˆt, y ˆµ ˆt therefore ˆµ(x ) ˆt and ˆµ(y ) ŝ ˆµ(f(x)) ˆt and ˆµ(f(y)) ŝ f 1 (ˆµ)(x) ˆt and f 1 (ˆµ)(y) ŝ xˆt f 1 (ˆµ) and yŝ f 1 (ˆµ) (x y)ˆt.ŝ qf 1 (ˆµ) [Since f 1 (ˆµ) is an (, q)-interval-valued fuzzy dot d-ideal of X.) f 1 (ˆµ)(x y) ˆt.ŝ or f 1 (ˆµ)(x y) + ˆt.ŝ ˆ1 ˆµ(f(x y)) ˆt.ŝ or ˆµ(f(x y)) + ˆt.ŝ ˆ1 ˆµ(f(x) f(y)) ˆt.ŝ or ˆµ(f(x) f(y)) + ˆt.ŝ ˆ1 (x y )ˆt.ŝ ˆµ or (x y )ˆt.ŝqˆµ (x y )ˆt.ŝ qˆµ x ˆt, y ŝ ˆµ (x y )ˆt.ŝ qˆµ Hence ˆµ is an (, q)-interval-valued fuzzy dot d-ideal of X. References [1] M. Akram and K. H. Dhar, on fuzzy d algebras, Punjab University Journal of Math, 37(005), [] N. O. Al-Shehrie, On fuzzy dot d ideals of d algebras, Advances in Algebra (ISSN ),(009), 1-8. [3] K. T. Atanassov, Operations over interval-valued fuzzy sets, Fuzzy sets and systems, 64 (1994),

15 Advanced Trends in Mathematics Vol [4] D. K. Basnet and L. B. Singh, (, q) - fuzzy ideal of BG-algebra, International Journal of Algebra, 5(15) (011), [5] S. K. Bhakat and P. Das, (, q)-fuzzy subgroup, Fuzzy sets and systems, 80 (1996), [6] S. K. Bhakat and P. Das, ( q)-level subset, Fuzzy sets and systems, 103(3) (1999), [7] Y. Imai and K. Iseki, On Axiom systems of Propositional calculi XIV, Proc, Japan Academy, 4 (1966)19-. [8] K. H. Kim, on fuzzy dot subalgebras of d-algebras, International Mathematical Forum,4(009), [9] D. S. Lee and C. H. Park, Interval-valued (, q) fuzzy ideals in Rings, International Mathematical Forum, 4(13) (009), [10] P. P. Ming and L. Y. Ming, Fuzzy topologyi, Neighbourhood structure of a fuzzy point and Moore- Smith convergence,j. Maths. Anal. Appl,76(1980), [11] J. Neggers and H. S. Kim, on d-algebras, Math.Slovaca, 49(1996),19-6. [1] O.G. Xi, Fuzzy BCK algebras, Math Japonica,36(1991), [13] Y. B. Yun, On (α, β)-fuzzy ideals of BCK/BCI-Algebras, Scientiae Mathematicae Japonicae, online, e-004, [14] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), [15] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-i, Information Science 8 (1975),

16 Volume / (, Vq)-Interval-Valued Fuzzy Dot d-ideals of d-algebras / DOI References [3] K. T. Atanassov, Operations over interval-valued fuzzy sets, Fuzzy sets and systems, 64 (1994), / (94)90331-x [5] S. K. Bhakat and P. Das, (, q)-fuzzy subgroup, Fuzzy sets and systems, 80 (1996), / (95) [6] S. K. Bhakat and P. Das, ( q)-level subset, Fuzzy sets and systems, 103(3) (1999), /s (97) [7] Y. Imai and K. Iseki, On Axiom systems of Propositional calculi XIV, Proc, Japan Academy, 4 (1966) /pja/ [14] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), /s (65)9041-x [15] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-i, Information Science 8 (1975), / (75)