Complete and Fuzzy Complete d s -Filter

Transcription

1 International Journal of Mathematical Analysis Vol. 11, 2017, no. 14, HIKARI Ltd, Complete and Fuzzy Complete d s -Filter Habeeb Kareem Abdullah University of Kufa Faculty of Education for Girls Department of Mathematics, Iraq Ghadeer Kareem Saeed University of Kufa Faculty of Education for Girls Department of Mathematics, Iraq Copyright 2017 Habeeb Kareem Abdullah and Ghadeer Kareem Saeed. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we will define a new filter on d s -algebra, is called c-d s -filter, and we study the relationship between him and the others filters. Also, we will define a new fuzzy filter on d s -algebra, is called fuzzy c-d s -filter and we study some of its characteristics. Mathematics Subject Classification: 08A72, 93E11, 97H99, Keywords: Filter, Implicative, Commutative, Involutory, Fuzzy filter 1 Introduction In 2014, A. Khaled introduced a d s -algebra and he defined d-filter in d s -algebra [1]. In 2016, H.Z.Ahmed introduced c-bck-ideal [5], and fuzzy c-bck-ideal [6]. K. Taher, in 2016 introduced c-bck-filter and fuzzy c-bck-filter [3]. In this paper, we will introduce a new concepts in d s -algebra and we study the relationship between these concepts. In the first item, we will show some preliminaries on which we base our paper. And also we will define the notions d s - filter and implicative filter, such that every implicative d s -filter is d s -filter, also we define fuzzy d s -filter. In the second item, we will show a new notion is called complete d s -filter and we study some of its characteristiscs and its relation to d s -

2 658 Habeeb Kareem Abdullah and Ghadeer Kareem Saeed filter and we prove that, every d s -filter is complete d s -filter, but the converse is not true in general. In the third item, we will define fuzzy complete d s -filter and we study some of its characteristics and we debate his relationship with fuzzy d s - filter, such that every fuzzy d s -filter is fuzzy complete d s -filter, but the converse is not true in general. 2 Preliminaries In this item, we will offer some of the concepts we need it in this paper, such as d s -algebra, d s -filter, fuzzy d s -filter, etc.. Definition (2.1):[1] A d s -algebra is a non-empty set X with a constant 0, and a binary operation, the symbol (X,,0), if it satisfies the following condition: i.x x = 0 ii.0 x = 0 iii.x y = 0 and y x = 0 imply that x = y iv.x 0 = x v.(x y) z = (x z) y. x, y, z X. We will refer to x y by xy. Remark (2.2): Let (X,,0) be a d s -algebra. In X we can define a relation by x y if and only if xy = 0. Definition (2.3): [1] A d s -algebra X is said to be bounded if there is an element e X such that x e for all x X, i. e. xe = 0, x X. Proposition (2.4): [1] In a bounded d s -algebra X the following properties are held i.(xy) x. ii.x y implies y x. iii.x(xy) y, x, y X. In bounded d s -algebra, we denote ex by x for every X. Definition (2.5): A d s -algebra X is said to be commutative if x(xy) = y(yx) for all x, y X, and y(yx) is denoted by (x y). Remark (2.6): In a bounded d s -algebra X, for all x, y X, we define (x y) = (x y ). Definition (2.7): A d S -algebra X is said to be implicative if satisfies the identity x(yx) = x, for all x, y X. Proposition (2.8): [1] In a bounded commutative d s -algebra X the following properties are held i.(x ) = x, x X. ii.x y = (x y) and x y = (x y). iii.x y = yx, x, y X. iv.(x y) y, x, y X. v. x 0 = x, x e = e, x, y X. Definition (2.9): A d s -algebra X in which (x ) = x for all elements is called involutory. Theorem (2.10): Let (X, 0) be a d s -algebra and e X. We define the operation on

3 Complete and fuzzy complete d s -filter 659 xy if x, y X, 0 if x X and y = e, X = X {e} as follows x y = { e if x = e and y X, 0 if x = y = e, Then (X, 0) is bounded d s -algebra with unit e and it's called the Iseki's extension of (X,,0). Proof. Since X is d s - algebra, keep to prove X {e} is d s -algebra, i. ee = 0 if x = y = e. ii. 0e = 0 if x = 0 X and y = e. iii. ee = 0 and ee = 0 imply e = e if x = y = e. iv. e0 = e if x = e and y = 0 X. v. (ey)z = (ez)y, (ee)z = (ez)e, (ey)e = (ee)y, (ee)e = (ee)e, (xe)z = (xz)e, (xy)e = (xe)y, (xe)z = (xz)e. This show that (X, 0) is a d s -algebra. And it is clear that e is unit of X. The proof is complete. Proposition (2.11): Let X be a d s -algebra and X be the Iseki's extension of X. Then x = 0 or x = e, for all x X. Proof. Since x = ex then by theorem (2.10), either x X so x = e, or x = e so x = 0. Definition (2.12): Let f be a mapping from a d s -algebra Xinto a d s -algebray. Then fis called 1-Homomorphism if f(x y) = f(x) f(y) for all x, y X. 2-Epimorphism if f is homomorphism and onto. 3-Monomorphism if f is homomorphism and one to one. 4-Isomorphism if f is epimorphism and monomorphism. 5-Isotone if x y f(x) f(y), for all x, y X. Lemma (2.13): If f is an epimorphism from d s -algebra X into d s -algebra Y, then f(e x ) = e y where e x, e y are the units of X and Y respectively. Proof. Since f is an epimorphism, there exists x X, such that f(x) = e y, clearly x e x, thus f(x) f(e x ) by (definition (2.12)(5)), that is e y f(e x ), on the other hand, f(e x ) e y, so f(e x ) = e y. Lemma (2.14): Let f is an isomorphism from d s -algebra X into d s -algebra, then 1-f(x ) = (f(x)), for all x X. 2-f 1 (y ) = (f 1 (y)), for all y Y. Proof. 1- f(x ) = f(e x x) = f(e x ) f(x) = e y f(x) = (f(x)) by (lemma (2.13)). 2- f 1 (y ) = f 1 (e y y) = f 1 (e y ) f 1 (y) = e f 1 (y) = (f 1 (y)) by (lemma (2.13)). Definition (2.15): A non-empty subset F of X is called d s -filter of X if (F 1 ) e F, (F 2 ) (x y ) F and y F imply x F for all x, y X.

4 660 Habeeb Kareem Abdullah and Ghadeer Kareem Saeed Definition (2.16): A non-empty subset F of X is called implicative d s -filter of d s - algebra X if, (F 1 ) e F (F 2 ) ((x (y x ))z ) F and z F imply x F, x, y, z X. Proposition (2.17): Every implicative d s -filter of d s -algebra X is d s -filter. Proof. Let (x y ) F and y F. Since (x (x x )) = x, then ((x (x x ))y ) F, y F, since F is an implicative d s -filter, then x F. Thus F is d s -filter. Definition (2.18): [4] Let X be a non-empty set. A fuzzy set μ in X is a function from X into the closed interval [0,1] of the real numbers. Definition (2.19): [2] Let μ and ν be two fuzzy sets in X. Then: i.(μ ν)(x) = min{μ(x),ν(x)}, for all x X. ii.(μ ν)(x) = max{μ(x),ν(x)}, for all x X. μ ν and μ ν are fuzzy sets in X. In general, if {μ, λ} is a family of fuzzy sets in X, then α λ μ α (x) inf{μ α (x), α λ}, for all x X and α λ μ α (x) sup{ μ α (x), α λ}, for all x X. Which are also fuzzy sets in X. Definition (2.20): [1] Let X be a bounded d s -algebra. Then for any fuzzy subsets μ and, the fuzzy subset μη is defined by μη(x) = sup x=y z {min{μ(y), η(z)}}. Definition (2.21): Let μ be a fuzzy set in a bounded d s -algebra X. Then μ is called a fuzzy d s -filter of X if (FF 1 ) μ(e) μ(x), (FF 2 ) μ(x) min {μ(x y ), μ(y)}, for all x, y X. Proposition (2.22): If μ is a fuzzy d s -filter in a bounded commutative d s -algebra X, then the following are held i.x y, imply μ(y) μ(x), ii. y x, then μ(y) μ(x), x, y X. Proof. i. From x y it follow that (x y ) = e, and so μ(x) min {μ((x y )), μ(y)} = min{μ(e), μ(y)} = μ(y). ii. Since y x then yx = 0, but x y = yx (by proposition (2.8)(iii)), i.e. x y and by (i), we get μ(y) μ(x). 3 Complete d s -filter In this item, we will define a new filter, we called complete d s -filter, and we study some of its properties. Definition (3.1): A non-empty subset Fof a d s -algebra X is called complete d s - filter or c-d s -filter for short, if 1- e F, 2-(x y ) F and y Fimply x F.

5 Complete and fuzzy complete d s -filter 661 Example (3.2): Let X = {0, a, b, c, d} and a binary operation is defined by 0 a b c d a a 0 a c 0 b b b c c 0 c 0 0 d d b a c 0 It's clear that (X,,0) is a bounded d s -algebra with unit d and F = {a, d} it's easy to show its c-d s -filter in X. Example (3.3): Let X = {0, a, b} and a binary operation is defined by 0 a b a a 0 0 b b b 0 It's clear that (X,,0) is a bounded d s algebra with unit b and F = {0, b} is not c-d s -filter since (a 0 ) = b F &(a b ) = 0 F but a F. Remark (3.4): In general {e}, X are trivial c-d s -filter. Proposition (3.5): In d s -algebra X every d s -filter is c-d s -filter. Proof. Let F be a d s -filter and let (x y ) F, y F. Since F is filter, then x F. Thus F is c-d s -filter. Remark (3.6): The converse of above proposition need not be true in general as in the following example. Example (3.7): Let X = {0, a, b} and a binary operation is defined by 0 a b a a 0 0 b b a 0 It's clear that (X,,0) is bounded d s -algebra with unit b, F = {a, b} is c-d s -filter, but it's not d s -filter, since (0 a ) = a F and a F but 0 F. Corollary (3.8): Every implicative d s -filter is c-d s -filter. Proposition (3.9): Every subset contained e in involutory d s -algebra X is c-d s - filter. Proof. It's clear by (x e ) = x for all x X. Example (3.10): In example (3.2) it's easy to show that (X, 0) is an involutory bounded d s -algebra with unit d and {d}, {0, d}, {a, d}, {b, d}, {0, a, d}, ect. are c-d s -filter. Corollary (3.11): Every subset contained e in bounded commutative d s -algebra X is c-d s -filter.

6 662 Habeeb Kareem Abdullah and Ghadeer Kareem Saeed Corollary (3.12): Every subset contained e in bounded implicative d s -algebra X is c-d s -filter. Proposition (3.13): Let X be a d s -algebra and X be the Iseki's extension of X. Then: 1-Every proper subset F of X contain 0 is not c-d s -filter. 2-Every proper subset F of X not contain 0 and contain e is c-d s -filter. Proof. 1-Let F be a proper subset of X and 0 F, then x X and x F such that (x y ) F, y F (by proposition (2.11)). 2-Since (x e ) = 0 F, x X. Remark (3.14): Note that the intersection and union of two c-d s -filter are not necessary to be c-d s -filters as shown in the following example. Example (3.15): Let X = {0, a, b, c, d} and a binary operation is defined by 0 a b c d a a 0 a 0 0 b b c c c c 0 0 d d c d a 0 Then (X,,0) is a bounded d s -algebra with unit d. Now let F 1 = {0, a, d} and F 2 = {0, c, d}, we can show easily that F 1 and F 2 are c-d s -filter in X. But F 1 F 2 = {0, d} can not be c-d s -filter, since (b 0 ) = d F 1 F 2 and (b d ) = 0 F 1 F 2, but b F 1 F 2. Also F 1 F 2 = {0, a, c, d} can not be c-d s -filter, since(b 0 ) = d F 1 F 2, (b a ) = c F 1 F 2, (b c ) = a F 1 F 2 and (b d ) = 0 F 1 F 2, but b F 1 F 2. Proposition (3.16): Let f be an isomorphism from a bounded d s -algebra X into a bounded d s -algebra Y. If F is c-d s -filter in X, then f(f) is c-d s -filter in Y. Proof. Let f be an isomorphism function from bounded d s -algebra X into bounded d s -algebra Y, and let F be a c-d s -filter in X. Then e x F. So f(e x ) = e y f(f), (by lemma (2.13)). Now let (x y ) f(f), y f(f), then f 1 ((x y ) ) F, f 1 (y) F, (since f is onto). But f 1 ((x y ) ) = ((f 1 (x) (f 1 (y) )), (by lemma (2.14), (2)). Therefore ((f 1 (x) (f 1 (y) )) F, f 1 (y) F. Since F is c-d s -filter in X, then f 1 (x) F, thus x f(f). That mean f(f) isc -d s -filter in Y. Proposition (3.17): Let f an be epimorphism from a bounded d s -algebra X into a bounded d s -algebra Y. If D is c-d s - filter in Y, then f 1 (D) is c-d s -filter in X. Proof. Let f be an epimorphism from a bounded d s -algebra X into a bounded d s - algebra Y, and let D be a c-d s -filter in Y, so e y D.

7 Complete and fuzzy complete d s -filter 663 Then f(e x ) = e y D, so e x f 1 (D), ( by lemma (2.13)). Now let (x y ) f 1 (D), y f 1 (D), so f((x y ) ) D, f(y) D, (since f is onto ). But f((x y ) ) = ((f(x)) (f(y)) ) D, f(y) D, since D is a c-d s -filter in Y, then f(x) D, therefore x f 1 (D). Thus f 1 (D) is a c-d s -filter in X. 4 Fuzzy complete d s -filter In this item, we will define a new filter, we called fuzzy complete d s -filter, and we study some of its properties. Definition (4.1): Let F be c-d s -filter of bounded d s -algebra. A fuzzy subset μ F is said to be fuzzy c-d s -filter if, 1-μ F (e) μ F (x), x X. 2-μ F (x) min{μ F ((x y ) ), μ F (y)}, y F. 3-If x y then μ F (x) μ F (y), for any x, y X. Example (4.2): In example (3.2), let μ F be the fuzzy set defined as the following 0.7 if x = c, d μ F (x) = { 0.3 if x = 0, a, b Then it's clear to show that μ F is fuzzy c-d s -filter in X. Example (4.3): Let X = {0, a, b, c, d, e} and a binary operation is defined by 0 a b c d e a a 0 a a 0 0 b b b c c c b 0 b 0 d d b a a 0 0 e e c d a b 0 It's clear that (X,,0) is a bounded involutory d s -algebra with unit e. Notice that F = {d, e} is c-d s -filter. Now we can show easily for a fuzzy c-d s -filter is μ F = { t 1 if x = a, d, e, where t 1 t 2 t 2 if x = 0, b, c 0.7 if x = e 0.5 if x = d But θ F (x) = { 0.4 if x = c 0.1 if x = 0, a, b Is not fuzzy c-d s -filter, since θ F (b) = 0.1 min{θ F ((b d ) ), θ F (d)} = 0.3. Remark (4.4): Every fuzzy constant in bounded d s -algebra X is fuzzy c-d s -filter. Proposition (4.5): Let F be c-d s -filter of bounded commutative d s -algebra. A fuzzy subset μ F is said to be fuzzy c-d s -filter if, 1-μ F (e) μ F (x), x X. 2-μ F (x) min{μ F ((x y ) ), μ F (y)}, y F. Proof: By proposition (2.22)

8 664 Habeeb Kareem Abdullah and Ghadeer Kareem Saeed Proposition (4.6): Every fuzzy d s -filter in bounded d s -algebra X is fuzzy c-d s - filter. Proof. Let F be c-d s -filter and let μ be a fuzzy d s -filter, then 1-μ(e) μ(x), x X. (by (FF 1 )in definition (2.21)). 2-μ(x) min{μ((x y ) ), μ(y)}, x, y X. Since F X. Then μ(x) min{μ((x y ) ), μ(y)}, y F. 3-If x y then μ(x) μ(y), for any x, y X (by proposition (2.22),(ii)). Thus μ is fuzzy c-d s -filter. Remark (4.7): The conversely of above proposition is not true in general as in the example (4.2). μ F is fuzzy c-d s -filter in X, but μ F is not fuzzy d s -filter, since; μ F (0) = 0.3 min{μ F ((0 c ) ), μ F (c)} = 0.7. Proposition (4.8): Let F be c-d s -filter of bounded involutory d s -algebra. A fuzzy subset μ F is fuzzy c-d s -filter if and only if satisfies 1-μ F (e) μ F (x), x X. 2-μ F (x) min{μ F ((yx) ), μ F (y)}, y F. 3-If x y then μ F (x) μ F (y), for any x, y X. Proof. It's clear. Corollary (4.9): Let F be c-d s -filter of bounded commutative d s -algebra. A fuzzy subset μ F is fuzzy c-d s -filter if and only if satisfies 1-μ F (e) μ F (x), x X. 2-μ F (x) min{μ F ((yx) ), μ F (y)}, y F. 3-If x y then μ F (x) μ F (y), for any x, y X. Lemma (4.10): Let F be c-d s -filter in bounded d s -algebra X. If {μ if : i } is a family of fuzzy c-d s -filters, then i μ if is fuzzy c-d s -filter. Proof. 1-Let x X, μ if (e) μ if (x), inf{μ if (e)} inf{μ if (x)}, so i μ if (e) i μ if (x). 2-Let x X, μ if (x) min{ μ if ((x y ) ), μ if (y)}, y F. Thus, inf{μ if (x)} inf{min{μ if ((x y ) ), μ if (y)}}, y F {min{inf μ if ((x y ) ), inf μ if (y)}}, y F, so i μ if (x) min{ i μ if ((x y ) ), i μ if (y)}, y F. 3-If x y, then μ if (x) μ if (y), for any x, y X, inf{μ if (x)} inf{μ if (y)}, for any x, y X, so i μ if (x) i μ if (y), for any x, y X. Then i μ if is fuzzy c-d s -filter. Remark (4.11): In bounded d s -algebra X. The union of two fuzzy c-d s -filters in general, it is not necessary fuzzy c-d s -filter, as it's shown in the following example. Example (4.12): Consider the involutory bounded d s -algebra X in example (4.3), F = {d, e}, and define μ F (x) = { 0.6 if x = c, e 0.4 if x = 0, a, b, d, η 0.7 if x = a, d, e F(x) = { 0.2 if x = 0, b, c Then it's clear to show that μ F, η F are fuzzy c-d s -filters. 0.7 if x = a, d, e But (μ F η F )(x) = { 0.6 if x = c 0.4 if x = 0, b

9 Complete and fuzzy complete d s -filter 665 Is not fuzzy c-d s -filter since (μ F η F )(b) = 0.4 min {((μ F η F )((d b) ), (μ F η F )(d)} = 0.6. Proposition (4.13): Let μ F, η F are two fuzzy c-d s -filter in a commutative bounded d s -algebra X, then μ F η F = μ F η F. Proof. Let z X, such that z = x y. Then, μ F (x) μ F (x y) = μ F (z) and η F (y) η F (x y) = η F (z), since μ F and η F are fuzzy c-d s -filters. Thus min{μ F (x), η F (y)} (μ F η F )(z), therefore μ F η F μ F η F. Conversely, since z = z z for all z X, then (μ F η F )(z) = min{μ F (z), η F (z)} sup z=x y {min{μ F (x), η F (y)} = (μ F η F )(z). So μ F η F μ F η F. Thus, μ F η F = μ F η F. Corollary (4.14): If μ F, η F are two fuzzy c-d s -filter in a commutative bounded d s -algebra X, then μ F η F is fuzzy c-d s -filter. References [1] A.K. Hasan, On Fuzzy Filter Spectrum of d-algebra, M.Sc. Thesis, Faculty of Education for Girls, University of Kufa, [2] D. Dubois and H. Prade, Fuzzy Sets and Systems, Academic Press Inc,, New York, [3] K.T. Radhi, On Zariski Topology of Fuzzy BCK-algebra, M.Sc. Thesis, Faculty of Education for Girl, University of Kufa, [4] L. A. Zadeh, Fuzzy set, Inform. and Control., 8 (1965), [5] H.Z. Ahmed, H.K. Abdullah, Complete BCK-Ideal, European Journal of Scientific Research, 137 (2016), [6] H.Z. Ahmed, Introduce Some New Types of Ideal and Spectrum of Fuzzy BCK-algebra, Thesis, Faculty of Education for Girls, University of Kufa, Received: June 26, 2017; Published: July 10, 2017