Intuitionistic Fuzzy Bi-Ideals of Ternary Semigroups

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1 International Mathematical Forum, Vol. 7, 2012, no. 8, Intuitionistic Fuzzy Bi-Ideals of Ternary Semigroups S. Lekkoksung Rajamangala University of Technology Isan Khon Kaen Campus, Thail Lekkoksung Abstract In this paper we consider the intuitionistic fuzzification of the concept of bi-ideal (1,2)-ideal in a ternary semigroup T, investigate some properties of such ideals. Mathematics Subject Classification: 06F35, 20M12, 04A72 Keywords: Ternary semigroup, intuitionistic fuzzy (1,2)-ideal, intuitionistic fuzzy bi-ideal, intuitionistic fuzzy ideal 1 Introduction In 1932, Lehmer gave the definition of ternary semigroups. A nonempty set T is called a ternary semigroup if there exists a ternary operation T T T T, written as (a, b, c) abc satisfying the following identity ( a, b, c, d, e T )(((abc)de) =(a(bcd)e) =(ab(cde)). Any semigroup can be reduced to a ternary semigroup but a ternary semigroup does not necessarily reduce to a semigroup. However, Banach shows that a ternary semigroup does not necessarily reduce to a semigroup by this example. Example 1.1 ([2]) T = { i, 0,i} is a ternary semigroup while T is not a semigroup under the multiplication over complex numbers. The next example is also a ternary semigroup but not a semigroup. Example 1.2 ([2]) Z is a ternary semigroup but not a semigroup under the multiplication over integers. 2 Preliminary Notes Let T be a ternary semigroup. By a ternary subsemigroup of T we mean a non-empty subset A of T such that AAA A, by a left (right) ideal of T we mean a non-empty subset A of T such that TTA A (AT T A). By

2 386 S. Lekkoksung two-sided ideal or simply ideal, we mean a non-empty subset of T which is both a left a right ideal of T. A ternary subsemigroup A of a ternary semigroup T is called a bi-ideal of T if ATTTA A. A ternary subsemigroup A of T is called a (1;2)-ideal of T if ATTTATA A. A ternary semigroup T is said to be regular if, for each x T, there exist s, t, y T such that x = xsytx. A ternary semigroup T is said to be completely regular if, for each x T, there exist s, t, y, z S such that x = xsytx xzy = yzx. By a fuzzy set μ in a non-emptyset T we mean a function μ : S [0; 1], the complement of μ, denoted by μ, is the fuzzy set in T given by μ(x) = 1 μ(x) for all x T. An intuitionistic fuzzy set (briefly, IFS) A in a non-empty set X is an object having the form A = {(x; μ A (x); γ A (x)) x X} where the functions μ A : X [0; 1] γ A : X [0; 1] denote the degree of membership the degree of nonmembership, respectively, 0 μ A (x)+γ A (x) 1 for all x X. An intuitionistic fuzzy set A = {(x; μ A (x); γ A (x)) x X} in X can be identified to an ordered pair (μ A ; γ A )ini X I X. For the sake of simplicity, we shall use the symbol A =(μ A ; γ A ) for the IFS A = {(x; μ A (x); γ A (x)) x X}. 3 Main Results In what follows, let T denote a ternary semigroup unless otherwise specified. Definition 3.1 An IFS A =(μ A ; γ A ) in T is called an intuitionistic fuzzy ternary subsemigroup of T if (i) μ A (xzy) min{μ A (x),μ A (y)}; (ii) γ A (xzy) max{γ A (x),γ A (y)}. for all x, y, z T. Definition 3.2 An IFS A =(μ A ; γ A ) in T is called an intuitionistic fuzzy left ideal of T if μ A (xzy) μ A (y) γ A (xzy) γ A (y) for all x, y, z T. An intuitionistic fuzzy right ideal of T is defined in an analogous way. An IFS A =(μ A ; γ A ) in T is called an intuitionistic fuzzy ideal of T if it is both an intuitionistic fuzzy right an intuitionistic fuzzy left ideal of T. It is clear that any intuitionistic fuzzy left (right) ideal of S is an intuitionistic fuzzy subsemigroup of S.

3 Intuitionistic fuzzy bi-ideals of ternary semigroups 387 Definition 3.3 An intuitionistic fuzzy ternary subsemigroup A =(μ A ; γ A ) of T is called an intuitionistic fuzzy bi-ideal of T if (i) μ A (xswty) min{μ A (x),μ A (y)}. (ii) γ A (xswty) max{γ A (x),γ A (y)}. for all w, x, y, s, t T. Theorem 3.4 If {A i } i λ is a family of intuitionistic fuzzy bi-ideals of T, then A i is an intuitionistic fuzzy bi-ideal of T, where A i =( μ Ai, γ Ai ) μ Ai (x) = inf{μ Ai (x) i λ, x T }, γ Ai (x) = sup{γ Ai (x) i λ, x T }. Proof. Let x, y, z T. Then we have μ Ai (xzy) {min{μ Ai (x),μ Ai (y)}} = min{min{μ Ai (x),μ Ai (y)}} = min{min{μ Ai (x)}, min{μ Ai (y)}} = min{ μ Ai (x), μ Ai (y)}, γ Ai (xzy) {max{γ Ai (x),γ Ai (y)}} = max{max{γ Ai (x),γ Ai (y)}} = max{max{γ Ai (x)}, max{γ Ai (y)}} = max{ γ Ai (x), γ Ai (y)}. Hence A i is an intuitionistic fuzzy ternary subsemigroup of T. Next for x, y, a, s, t T, we obtain μ Ai (xsaty) {min{μ Ai (x),μ Ai (y)}} = min{min{μ Ai (x),μ Ai (y)}} = min{min{μ Ai (x)}, min{μ Ai (y)}} = min{ μ Ai (x), μ Ai (y)}, γ Ai (xzy) {max{γ Ai (x),γ Ai (y)}} = max{max{γ Ai (x),γ Ai (y)}} = max{max{γ Ai (x)}, max{γ Ai (y)}} = max{ γ Ai (x), γ Ai (y)}. Hence A i is an intuitionistic fuzzy bi-ideal of T. This completes the proof. Theorem 3.5 If an IFS A =(μ A ; γ A ) in T is an intuitionistic fuzzy bi-ideal of T, then so is A := (μ A, μ A ). Proof. It is sufficient to show that μ A sa tisfies the condition (ii) in Definition 3.1, (ii) in Definition 3.3. For any a, x, y, z, s, t T, we have μ A (xzy) = 1 μ A (xzy) 1 min{μ A (x),μ A (y)} = max{1 μ A (x), 1 μ A (y)} = max{ μ A (x), μ a (y)} μ A (xsaty) =1 μ A (xsaty) 1 min{μ A (x),μ A (y)} = max{1 μ A (x), 1 μ A (y)} = max{ μ A (x), μ A (y)}. Therefore A is an intuitionistic fuzzy bi-ideal of T.

4 388 S. Lekkoksung Definition 3.6 An intuitionistic fuzzy ternary subsemigruop A =(μ A,γ A ) of T is called an intuitionistic fuzzy (1,2)-ideal of T if (i) μ A (xawb(ycz)) min{μ A (x),μ A (y),μ A (z)}, (ii) γ A (xawb(ycz)) max{γ A (x),γ A (y),γ A (z)}, for all w, x, y, z, a, b, c T. Theorem 3.7 Every intuitionistic fuzzy bi-ideal is an intuitionistic fuzzy (1,2)-ideal. Proof. Let A = (μ A,γ A ) be an intuitionistic fuzzy bi-ideal of T w, x, y, z, a, b, c T. Then μ A (xawb(ycz)) = μ A ((xawby)cz) min{μ A (xawby),μ A (z)}} min{min{μ A (x),μ A (y)},μ A (z)} = min{μ A (x),μ A (y),μ A (z)}, γa (xawb(ycz)) = γ A ((xawby)cz) max{γ A (xawby),γ A (z)}} max{max{γ A (x),γ A (y)},γ A (z)} = max{γ A (x),γ A (y),γ A (z)}. Hence A =(μ A,γ a ) is an intuitionistic fuzzy (1,2)-ideal of T. To consider the converse of Theorem 3.8, we need to strengthen the condition of a ternary semigroup T. Theorem 3.8 If T is a regular ternary semigroup, then every intuitionistic fuzzy (1,2)-ideal of T is an intuitionistic fuzzy bi-ideal of T. Proof. Assume that a ternary semigroup T is regular let A = (μ A,γ A ) be an intuitionistic fuzzy (1,2)-ideal of T. Let w, x, y, s, t T. since T is regular, we have xsw (xtttx)tt xtttx which implies that xsw = xpsqx for some p, q T. Thus μ A (xswty) = μ A ((xpsqx)ty) =μ A (xpsq(xty)) min{μ A (x),μ A (x),μ A (y)} = min{μ A (x),μ A (y)}, γa (xswty) = γ A ((xpsqx)ty) =γ A (xpsq(xty)) max{γ A (x),γ A (x),γ A (y)} = max{γ A (x),γ A (y)}, Therefore A =(μ A,γ A ) is an intuitionistic fuzzy bi-ideal of T. Theorem 3.9 Let A =(μ A,γ A ) is an intuitionistic fuzzy bi-ideal of T. If T is a completely regular, then A(a) =A(a 3 ) for all a T.

5 Intuitionistic fuzzy bi-ideals of ternary semigroups 389 Proof. Let a T. Then there exist x, s, t T such that a = a 3 sxta 3. Hence μa (a) = μ A (a 3 sxta 3 ) min{μ A (a 3 ),μ A (a 3 )} = μ A (a 3 ) min{μ A (a),μ A (a)} = μ A (a), γa (a) = γ A (a 3 sxta 3 ) max{γ A (a 3 ),γ A (a 3 )} = γ A (a 3 ) max{γ A (a),γ A (a)} = γ A (a). It follows that μ A (a) =μ A (a 3 ) γ A (a) =γ A (a 3 ) so that A(a) =a(a 3 ). References [1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets Systems 20 (1986), [2] R. Chinram, S. Saelee, Fuzzy Ideals Fuzzy Filters of Ordered Ternary Semigroups, Journal of Mathematics Research. Vol 2, No.1 (2010), [3] V. N. Dixit, S. Dewan, S, A note on quasi bi-ideals in ternary semigroups, International Journal of Mathematics Mathematical Sciences., 18 (1995), [4] A. Iampan. Lateral ideals of ternary semigroups, Ukrainian Math. Bull., 4 (2007), [5] K.H. Kim, J.G. Lee, On intuitionistic fuzzy bi-ideals of semigroups, Turk J Math., 29 (2005), [6] N. Kuroki, On fuzzy ideals fuzzy bi-ideals in semigroups, Fuzzy Sets Systems, 5 (1981), [7] N. Kuroki, On fuzzy semigroups, Inf. Sci., 53(1991), [8] S. Lajos, (1,2)-ideal characterizations of unions of group, Math. Seminar Notes (presently, Kobe J. Math) 5 (1971), Received: August, 2011