Anti fuzzy ideals of ordered semigroups

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1 International Research Journal of Applied and Basic Sciences 2014 Available online at ISSN X / Vol, 8 (1): Science Explorer Publications Anti fuzzy ideals of ordered semigroups Mohsen Asghari-Larimi 1 and Alieh Mirzaei 2 1. Department of Mathematics, Golestan University, Gorgan, Iran 2. Department of Mathematics,Faculty of Sciences, GhaemsharBranch,Islamic Azad University,Ghaemshar, Iran Corresponding Author asghari2004@yahoo.com ABSTRACT:In this paper we introduce the notion of anti ordered fuzzy points of an ordered semigroup S, and give a characterization of anti fuzzy left (resp. right) ideals of orderedsemigroup S. We also introduce the concepts ofanti product on two fuzzy setofan ordered semigroup S. Furthermore, we introduce and investigate the anti strongly convex of a fuzzy subset and we establish some characterization theorems. Mathematics Subject Classification:20M12 Keywords:Fuzzy sets; Ordered semigroup; Anti ordered fuzzy point; anti fuzzy left (right) ideal of an ordered semigroup; Prime fuzzy ideal. INTRODUCTION Given a set H, a fuzzy subset of H (or a fuzzy set in H) is, by definition, an arbitrary mapping µ:h [0,1] where [0,1] is the usual interval of real numbers. This important concept of a fuzzy set has been introduced by Zadeh in Since then, many papers on fuzzy sets appeared showing the importance of the concept and its applications (cf., for example, Mordeson and al., 2005, Kehayopuluand al., 1990). The study of the fuzzy algebraic structures has started in the pioneering paper of Rosenfeld in Rosenfeld introduced the notion of fuzzy groups and showed that many results in groups can be extended in an elementary manner to develop the theory of fuzzy group. Since then the literature of various fuzzy algebraic concepts has been growing very rapidly. For example, the concept of a fuzzy ideal of a semigroup was introduced by Kuroki in Liu in 1982, and Mukherjee and Sen in 1987, introduced and examined the notion of a fuzzy ideal of a ring.xie and Tang in 2008,defined fuzzy radicals and prime fuzzy ideals of ordered semigroups. This paper is structured as follows. After the introduction, in section 2, we recall some basic notions and results on hypergroups and fuzzy sets. In section 3, we study some properties of the concept of an anti ordered fuzzy point of an ordered semigroup S, and characterizes anti fuzzy ideals (resp. right, left ideals) generatedbyanti fuzzy ordered points of an ordered semigroup and we establish some characterization theorems. 2 Preliminaries Let X be a non-empty set. A mapping : X 0,1 is called a fuzzy setin X. We denote byf(x) the set all fuzzy subsets of X.A fuzzy subset is non-empty if f is not the constant map which assumes the values 0. Also, the mapping 1 F :H [0,1] x 1 F (x):= 1 is the greatest element of F(H), and the mapping 0 F :H [0,1] x 0 F (x):= 0 is the zero element of F(H) for all x H, i.e., 0 F µ 1 F. For any two fuzzy sets and of X, means that, for all ( x) ( x. The symbols, and defined by x X, ) ( )( x) min{ ( x), ( x)} and ( )( x) max{ ( x), ( x)} for all x X. A fuzzy set of R of the form t( 0) if y x, ( y) 0 if y x, 2

2 is said to be a fuzzy point with support x and value t and is denoted by xt. Let be a fuzzy set of R and xt be a fuzzy point. Definition 2.1. Let (S,., ) be an orderersemigroup. For x S, we define A {(y, z) S S x yz}. For A, B S, we denote AB {ab a A, b B}. Definition 2.2. A nonempty subset A of an ordered semigroups is called a left (resp. right) ideal of S if (1) SA A(resp. AS A).and (2) If a A and S b a, then b A. If A is both a left and a right ideal of S, then it is called a (two-sided) ideal of S. Anti fuzzy left (resp. right) ideals of ordered semigroups This section introduces the concept of an anti ordered fuzzy point of an ordered semigroup S, and characterizes anti fuzzy ideals (resp. right, left ideals) generated by anti fuzzy ordered points of an ordered semigroup. Some results are useful in sequal. Definition 3.1. Let (S,., ) be an ordered semigroup. The anti productf g is defined by (f g)(x) = (, ) [max{f(y), g(z)}], ifa 1 0, ifa 1, for all x S. Lemma 3.2.If f, g F(S), f g, and h F(S), then f h g h, h f h g. The proof is straightforward verification. Let S be an ordered semigroup. For H S, we define [H) {t S h tfor some h H}. For H = {a}, we write [a) instead of [{a}). Lemma 3.3. Let S be an ordered semigroup. Then, we have (1) [A) A A S. (2) If A B S, then[a) [B). (3) [A)[B) [AB) A, B S. (4) [A) = [[A)) A S. (5) [A)[B) = [AB) A, B S. The proofs are straightforward verification. Definition 3.4.A nonempty subset A of an ordered semigroups is called an anti left (resp. right) ideal of S if (1) SA A(resp. AS A). (2) If a A and a b S, then b A. If A is both an anti left and an anti right ideal of S, then it is called an anti (two-sided) ideal ofs. Definition 3.5. Let S be an ordered semigroup. A fuzzy subset f of S is called an anti fuzzy left ideal of S if (1) x y f(x) f(y). (2) f(xy) f(y) x, y S. A fuzzy subset f of S is called an anti fuzzy right ideal of S if (1) x y f(x) f(y). (2) f(xy) f(y) x, y S. A fuzzy subset f of S is called an anti fuzzy ideal of S if it is both an anti fuzzy left and an anti fuzzy right ideal of S. Definition 3.6. Let f be any fuzzy subset of an ordered semigroups. The set L(f, t) {x S f(x) t},wheret [0,1] is called a lower t-level cutof f. Lemma 3.7. Let S be an ordered semigroupand f a fuzzy subset of S. Then f is an anti fuzzy ideal of S if and only if the lower t-level cut L(f, t)(t (0,1]) of f is an ideal of S for L(f, t). Let f be an anti fuzzy ideal of S. If x L(f, t), then f(x) t. Since f is an anti fuzzy ideal of S, we have f(xy) f(x) t and f(yx) f(x) t for all y S. Thus xy L(f, t)and yx L(f, t). Furthermore, let x L(f, t), S y x. Then f(x) t. Since f is an anti fuzzy ideal of S, we have f(y) f(x) t, so y L(f, t). Therefore, L(f, t) is an ideal of S. For any x, y S, let t = f(x). Then x L(f, t). Since L(f, t) is an ideal of S, we have xy L(f, t), and so f(xy) t = f(x), for all y S. Let x yand λ = f(y). Then y L(f, λ). Since L(f, λ) is an ideal of S, we have x L(f, λ). Then f(x) λ = f(y). Therefore, f is an anti fuzzy right ideal of S. In a similar way we can show that f is also an anti fuzzy left ideal of S, and so f is an anti fuzzy ideal of S. 22

3 Definition 3.8. Let S be an ordered semigroup, a Sand λ [0,1]. An anti ordered fuzzy pointa of S is defined by the rule that a (x) = 1, if x [a), λ, if x [a). For all x S. It is accepted that a is a mapping from Sinto [0, 1], then an ordered fuzzy point of S is a fuzzy subset of S. For any fuzzy subset f of S, we also denote a f by a f in sequel. Definition 3.9.Let f be a fuzzy subset of S, we define [f)by the rule that [f)(x) = for all x S. A fuzzy subset of S is called antistrongly convex if f = [f). Theorem 3.10.Let fbe a fuzzy subset of an ordered semigroups. The f is an anti strongly convex fuzzy subset of S if and only if x y f(x) f(y) for all x, y S. Let x, y Sand x y. Then, by hypothesis we have f(x) = [f)(x) = f(w) f(w) f(y). Since [f)(x) = f(w)and x x, we have [f)(x) f(x) for all x S. Then [f) f. Conversely, for any x S, since [f)(x) = f(y) and y x, by hypothesis we have f(x) f(y), for all y S. Then f(x) [f)(x). This f [f). So f = [f). Proposition 3.11.Let f be an anti strongly convex fuzzy subset of an ordered semigroups. Then y λ f. Suppose that y λ is an ordered fuzzy point such that y λ f. Since y λ (x) f(x), for all x S, we have y λ (x) = y λ (x) f(x) = f(x). Thus y λ f. By propostion 3.4, the following corollary is obvious. Corollary 3.12.Let f be a fuzzy subset of an ordered semigroup S. Then [ ) y [f). Proposition 3.13.Let fand g be fuzzy subsets of S. Then the following statement are true: (1) For all f F(S), [f) f. (2) If f g, then[f) [g). (3) For all f, g F(S), [f g) [f) [g). (4) For all f F(S), [f) = [f). (5) For any anti fuzzy ideal f of S, f = [f). (6) If a is an anti ordered fuzzy point of S, then a = [a ). (1) For each x S, since [f)(x) = f(y), and x x, we have [f)(x) f(x). Then [f) f. (2) If f g, then for each x S, f(x) g(x). Thus [f)(x) = f(y) g(y) = [g)(x). Therefore, [f) [g). (3) Let x S. If A, then it is obvious that [f) [g) (x) = 1 [f g)(x). If A, then there exist, y, x S such that x yz, and [f) [g) (x) = {[f)(y) [g)(z)} = f(s) g(t) =, {f(s) g(t)} {f(s) g(t)} = {(f g)(yz)} = [f g)(x). 23

4 Thus [f g) [f) [g)for all f, g F(S). (4) If x, y Sand x y, then [f)(y) = f(y) f(y) = [f)(x). Thus, by Theorem 3.10, [f) is strongly convex and [f) = [f). (5) By Definition 3.5 and Theorem 3.10, if f is an anti fuzzy ideal of S, then f is an anti strongly convex fuzzy subset of S. By definition 3.9, f = [f). (6) Since each ordered fuzzy point a is a fuzzy subset of S, we have [a ) a by (1). On the other hand, for each x S, [a )(x) = a (y). The two cases are considered: Let x [a). Then a (x) = 1. Since a (y) 1 for all y S, we have [a )(x) = a (y) 1 = a (x). Let x [a). Then y [a) for all y x. Thus [a )(x) = λ = a (x). Definition Let S be an ordered semigroup, A Sand λ [0,1). Let λf be a fuzzy subset of S defined as follows: λf (x) = 1, if x A, λ, if x A. For all x S. If A = [a), thenλf = a. Lemma Let A,B be any nonempty subsets of an ordered semigroups. Then for any λ [0,1) the following statement are true: (1) λf λf = λf. (2) λf λf = λf. (3) λf ( ] = a. (4) If A is an anti ideal (right ideal, left ideal) of S, then λf is an anti fuzzy ideal (fuzzy right ideal, fuzzy left ideal) of S. (1) Let x Sand x A B. Then λf (x) =1. Since x Aand x B, we have λf (x) = λf (x) = 1, and so (λf λf )(x) = λf (x) λf (x) = 1. Let x A B. Then λf (x) = λ. Suppose that x A.Then λf (x) = λ. Since λf (x) λ, we have (λf λf )(x) = λf (x) λf (x) = λ. It implies that λf λf = λf. (2) Let x Sand x A B. Then λf (x) = 1. Suppose that x Aand x B, we have λf (x) = λf (x) = 1, and so (λf λf )(x) = λf (x) λf (x) = 1. Also, if x Aand x B, then λf (x) = 1 and λf (x) = λ, so (λf λf )(x) = 1. (x) = λ. Since x Aand x B. Then λf (x) = λand λf (x) = λ, and so Let x A B. Then λf (λf λf )(x) = λf (x) λf (x) = λ. It implies that λf λf = λf. (3) Let x [A). Then there exists y A such that y x. Thus ( a )(x) = a (x) y (x) = 1. Since a (x) 1 for any ordered fuzzy point a of S, we have ( a )(x) = a (x) 1. Then ( a )(x) = 1 = λf [ ) (x). If x [A), then λf [ ) (x) = λ. Since x [A), we have x [a) for all a A, and so a λ (x) = λ. It implies that ( a λ )(x) = a λ (x) = λ = λf [ ) (x). (4) Suppose that A is an anti left ideal of S. Let y Sand y A. Then λf (y) = λ. Since λf is a fuzzy subset of S, we have λf (xy) λ = λf (y) for all x S. If y A, then λf (y) = 1. Since A is an anti left ideal of S and x S, y A, we have xy A. Thus, λf (xy) = 1 = λf (y). Therefore λf (xy) λf (y) for all x, y S. On the other hand, ifx y and x A, then λf (x) = λ. Since λf is a fuzzy subset of S, we have λf (x) = λ λf (y). If x Athen λf (x) = 1. Since A is an anti left ideal of S and x y, we have y A, and so λf (x) = 1 = λf (y), which can be shown that λf is An anti fuzzy left ideal of S. Similarly, it can be shown that λf is an anti fuzzy ideal (right ideal) of S if A is an anti ideal (right ideal) of S. 24

5 ACKNOWLEDGEMENT The author is highly grateful to the referees for their constructive suggestions for improving the paper. REFERENCES Kehayopulu N, Tsingelis M On weakly prime ideals of ordered semigroups, Math. Japan Kuroki N On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5, Liu WJ Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8, Mordeson JN, Bhutani KR, Rosenfeld A Fuzzy Group Theory, World Scientific, Singapore, Mukherjee TK, Sen MK.1987.On fuzzy ideals of a ring, Fuzzy Sets and Systems 21, Rosenfield A Fuzzy groups, J. Math. Appl. 35, Sen MK On Γ-semigroups, Proceedings of the International Conference on Algebra and its application. Decker Publication, New York, 301, Xie XY, Tang J.2008.Fuzzy radicals and prime fuzzy ideals of ordered semigroups, Information Sciences Zadeh LA Fuzzy Sets, Inform and Control 8,