Ordered semigroups characterized by their (2; 2 _q)-fuzzy bi-ideals Young Bae Jun and Asghar Khan Department of Mathematics Education Gyeongsang National University Chinju 660-701, Korea e-mail:
[email protected],
[email protected] Muhammad Shabir Department of Mathematics Quaid-i-Azam University Islamabad, Pakistan e-mail:
[email protected] December 7, 2008
Abstract Using the idea of a quasi-coincedence of a fuzzy point with a fuzzy set, the concept of an ( ; )-fuzzy bi-ideal in ordered semigroups is introduced, which is a generalization of the concept of a fuzzy bi-ideal in ordered semigroups and some interesting characterizations theorems are obtained. A special attention is given to (2; 2 _q)-fuzzy bi-ideals.
AMS Mathematics Subject Classi…cation: 06F05, 20M12, 08A72 Keywords: Fuzzy algebra; Belong to; Quasi-coincident with; ( ; )-fuzzy bi-ideal
1
Introduction
The fundamental concept of a fuzzy set, introduced by L. A. Zadeh, provides a natural frame-work for generalizing several basic notions of algebra. The study of fuzzy sets in semigroups was introduced by Kuroki [19 21]. A systematic I would like to thank the Higher Education Commission of Pakistan for …nancial support under grant no. I-8/HEC/HRD/2007/182.
1
exposition of fuzzy semigroups was given by Mordeson et al., where one can …nd theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy …nite state machines and fuzzy languages. The monograph by Mordeson and Malik deals with the application of fuzzy approach to the concepts of automata and formal languages. Murali [25] proposed the de…nition of a fuzzy point belonging to a fuzzy subset under a natural equivalence on fuzzy subset. The idea of quasicoincidence of a fuzzy point with a fuzzy set, played a vital role to generate some di¤erent types of fuzzy subgroups. Bhakat and Das [2; 3] gave the concepts of ( ; )-fuzzy subgroups by using the "belongs to" relation (2) and "quasicoincident with" relation (q) between a fuzzy point and a fuzzy subgroup, and introduced the concept of an (2; 2 _q)-fuzzy subgroup. In [4] ; (2; 2 _q)-fuzzy subrings and ideals are de…ned. In [5] Davvaz de…ne (2; 2 _q)-fuzzy subnearring and ideals of a near ring. In [1] Bhakat de…ne (2 _q)-level subset of a fuzzy set. In [6] Jun and Song initiated the study of ( ; )-fuzzy interior ideals of a semigroup. In [7] Kazanci and Yamak study (2; 2 _q)-fuzzy bi-ideals of a semigroup. The concept of a fuzzy generalized …lters was introduced by Ma et al. in [22] and characterized R0 -algebras in terms of this notion. Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoritical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces and the like. This provides su¢ cient motivation to researchers to review various concepts and results from the realm of abstract algebra in broader framework of fuzzy setting. Our aim in this paper is to introduce and study the new sort of fuzzy bi-ideals called ( ; )-fuzzy bi-ideals and to study some interesting characterizations of ordered semigroups in terms of ( ; )-fuzzy bi-ideals. Special concentration is given to (2; 2 _q)-fuzzy bi-ideals and some characterizations of regular and intra-regular ordered semigroups are obtained by using (2; 2 _q)-fuzzy bi-ideals.
2
Preliminaries
An ordered semigroup is an ordered set S at the same time a semigroup such that a; b 2 S; a b =) xa xb and ax bx for all x 2 S. Let (S; :; ) be an ordered semigroup. For A S, we denote (A] := ft 2 Sjt
h for some h 2 Ag:
For A; B S; we denote, AB := fabja 2 A; b 2 Bg: Let A; B S. Then (A], (A](B] (AB], and ((A]] = (A]. Let S be an ordered semigroup and ; = 6 A S. Then A is called a subsemigroup of S if A2 A. A subsemigroup A of an ordered semigroup S is called a bi-ideal of S if (1) ASA A and (2) (8x 2 S)(8y 2 A) (x y =) x 2 A)[9]. A
De…nition 1 (cf.[13]) An element z of an ordered semigroup S is called left (resp. right) zero if for all x 2 S, zx = x (resp. xz = z). An ordered semigroup S is called left(resp. right) zero if every element of S is left(resp. right) zero. 2
An ordered semigroup S is called regular if for every a 2 S, there exists x 2 S such that a axa. Equivalent de…nitions:[15; 16] (1) (8a (2) (8A
2
S)(a 2 (aSa]). S)(A (ASA]).
An ordered semigroup S is called intra-regular if for every a 2 S there exist x; y 2 S such that a xa2 y. Equivalent de…nitions:[16] (1) (8a (2) (8A
2
S)(a 2 (Sa2 S]): S)(A (SA2 S]).
Let S be an ordered semigroup by a fuzzy subset A of S, we mean a function A : S ! [0; 1]. Let A be a fuzzy subset of S, then A is called a fuzzy subsemigroup [9] of S if (8x; y 2 S)(A(xy) minfA(x); A(y)g): De…nition 2 (cf. [9]) Let S be an ordered semigroup and A a fuzzy subset of S. Then A is called a fuzzy bi-ideal of S if: (B1 ) (8x; y 2 S) (x y =) A(x) A(y)). (B2 ) (8x; y 2 S) (A(xy) minfA(x); A(y)g): (B3 ) (8x; y; z 2 S)(A(xyz) fA(x); A(z)g). Let S be an ordered semigroup and A a fuzzy subset of S, then for all t 2 (0; 1], the set U (A; t) := fx 2 SjA(x) tg is called a level subset of S. Theorem 3 Let (S; :; ) be an ordered semigroup and A a fuzzy subset of S. Then A is a fuzzy bi-ideal of S if and only if the level subset U (A; t)(6= ;) is a bi-ideal of S for all t 2 (0; 1]. Let S be an ordered semigroup and ; = 6 B function B of B is de…ned as follows: B
: S ! [0; 1]jx !
B (x)
:=
S. Then the characteristic 1 if x 2 B, 0 if x 2 = B:
Lemma 4 (cf. [9, Theorem 1]). Let S be an ordered semigroup and B a nonempty subset of S. Then B is a bi-ideal of S if and only if B is a fuzzy bi-ideal of S.
3
For a 2 S; de…ne Aa := f(y; z) 2 S
Sja
yzg[10]:
For fuzzy subsets A1 and A2 of S, de…ne
A1 A2 : S ! [0; 1]ja !
8
1: It follows that 1 < A(x) + t A(x) + A(x) = 2A(x); so that A(x) > 0:5. This means that fxt jxt 2 ^qAg = ;: De…nition 7 A fuzzy subset A of S is called an ( ; )-fuzzy bi-ideal of S, where 6=2 ^q; if it satis…es: (B7 ) (8x; y 2 S)(8t 2 (0; 1])(x y, yt A =) xt A). (B8 ) (8x; y 2 S)(8t; r 2 (0; 1])(xt ; yr A =) (xy)m inft;rg A). (B9 ) (8x; y; z 2 S)(8t; r 2 (0; 1])(xt ; zr A =) (xyz)m inft;rg A). Example 8 Consider a set S = fa; b; c; d; eg with the following multiplication ":" and order relation " ": : a b c d e
a a a a a a
b d b d d d
c a a c a c
5
d d d d d d
e d d e d e
:= f(a; a); (a; c); (a; d); (a; e); (b; b); (b; d); (b; e); (c; c); (c; e); (d; d); (d; e); (e; e)g Then (S; :; ) is an ordered semigroup (see[17]) and fag, fa; b; eg and fa; b; d; eg are bi-ideals of S. We de…ne a fuzzy subset A : S ! [0; 1] by: A(a) = 0:8; Then
A(b) = 0:7;
A(e) = 0:6;
A(d) = 0:5;
A(c) = 0:3:
8 S if t 2 (0; 0:3]; > > > > < fa; b; d; eg if t 2 (0:3; 0:5]; fa; b; eg if t 2 (0:5; 0:6]; U (A; t) := > > fag if t 2 (0:6; 0:8]; > > : ; if t 2 (0:8; 1]
Clearly A is an (2; 2 _q)-fuzzy bi-ideal of S. But (i) A is not an (2; 2)-fuzzy bi-ideal of S, since a0:78 2 A and b0:66 2 A but (ab)m inf0:78;0:76g = d0:76 2A: (ii) A is not a (q,2)-fuzzy bi-ideal of S, since a0:75 qA and b0:65 qA but (ab)m inf0:75;0:65g = d0:65 2A: (iii) A is not an (2,q)-fuzzy bi-ideal of S, since a0:30 2 A and b0:20 2 A but (ab)m inf0:30;0:20g = d0:20 qA: (iv) A is not an (q; 2 _q)-fuzzy bi-ideal of S since a0:65 qA and b0:55 qA but (ab)m inf0:65;0:55g = d0:55 2 _qA: (v) A is not a (q,2 ^q)-fuzzy bi-ideal of S, since a0:72 qA and b0:62 qA but (ab)m inf0:72;0:62g = d0:62 2 ^qA: (vi) A is not an (2 _q,2 ^q)-fuzzy bi-ideal of S, since a0:64 2 _qA and b0:54 2 _qA but (ab)m inf0:64;0:54g = d0:54 2A and so d0:54 2 ^qA: (vii) A is not an (2 _q,2)-fuzzy bi-ideal of S, since a0:63 2 _qA and b0:53 2 _qA but (ab)m inf0:63;0:53g = d0:53 2A: (viii) A is not an (2; 2 ^q)-fuzzy bi-ideal of S, since a0:62 2 A and b0:52 2 A but (ab)m inf0:62;0:52g = d0:52 2A and so d0:52 2 ^qA: (xi) A is not a (q,q)-fuzzy bi-ideal of S, since a0:38 qA and b0:48 qA but (ab)m inf0:38;0:48g = d0:38 qA: 6
(x) A is not an (2 _q, q)-fuzzy bi-ideal of S, since a0:39 2 _qA and b0:49 2 _qA but (ab)m inf0:39;0:49g = d0:39 qA: (xi) A is not an (2 _q, 2 _q)-fuzzy bi-ideal of S, a0:68 2 _qA and b0:58 2 _qA but (ab)m inf0:68;0:58g = d0:58 2 _qA: Theorem 9 Every (2; 2)-fuzzy bi-ideal is an (2; 2 _q)-fuzzy bi-ideal. Proof. Straightforward. Theorem 10 Every (2 _q; 2 _q)-fuzzy bi-ideal is (2; 2 _q)-fuzzy bi-ideal. Proof. Let A be an (2 _q,2 _q)-fuzzy bi-ideal of S: Let x; y 2 S, x y and t 2 (0; 1] be such that yt 2 A: Then yt 2 _qA: Since x y and yt 2 _qA we have xt 2 _qA: Let x; y 2 S and t; r 2 (0; 1] be such that xt ; yr 2 A: Then xt ; yr 2 _qA; which implies (xy)m inft;rg 2 _qA: Let now, x; y; z 2 S and t; r 2 (0; 1] be such that xt ; zr 2 A. Then xt ; zr 2 _qA; which implies (xyz)m inft;rg 2 _qA: Theorem 11 Let A be a non-zero ( ; )-fuzzy bi-ideal of S. Then the set A0 := fx 2 SjA(x) > 0g is a bi-ideal of S. Proof. Let x; y 2 S; x y: If y 2 A0 ; then A(y) > 0. Since x y we have A(x) A(y); then A(x) > 0 and so x 2 A0 . Let x; y 2 A0 : Then A(x) > 0 and A(y) > 0. Assume that A(xy) = 0. If 2 f2; 2 _qg then xA(x) A and yA(y) A but (xy)m infA(x);A(y)g A for every 2 f2;q; 2 _q,2 ^qg; a contradiction. Note that x1 qA and y1 qA but (xy)m inf1;1g = (xy)1 A for every 2 f2;q; 2 _q,2 ^qg, a contradiction. Hence A(xy) > 0, that is xy 2 A0 . Now, let x; z 2 A0 and y 2 S. Then A(x) > 0 and A(z) > 0: Assume that A(xyz) = 0: If 2 f2; 2 _qg then xA(x) A and zA(z) A but (xyz)m infA(x);A(z)g A for every 2 f2;q; 2 _q,2 ^qg; a contradiction. Note that x1 qA and z1 qA but (xyz)m inf1;1g = (xyz)1 A for every 2 f2;q; 2 _q,2 ^qg, a contradiction. Hence A(xyz) > 0, that is xyz 2 A0 . Consequently, A0 is a bi-ideal of S. Theorem 12 Let S be a left(resp. right) zero ordered semigroup and A a nonzero (q,q)-fuzzy bi-ideal of S. Then A is constant on A0 . _ Proof. Let a be an element of S such that A(a) = fA(x)jx 2 Sg: Then a 2 A0 . Suppose that there exists x 2 A0 such that tx = A(x) 6= A(a) = ta : Then tx < ta . Choose r; s 2 (0; 1] such that 1 ta < r < 1 tx < s: Then ar qA and xs qA but (ax)m infr;sg = xr qA(resp. (xa)m infr;sg = xr qA); since S is right(resp. left) zero, a contradiction. Hence A(x) = A(z) for all x 2 A0 . Theorem 13 Let B be a bi-ideal and A a fuzzy subset of S such that
7
(1) (8x 2 SnB) (A(x) = 0) (2) (8x 2 B) (A(x) 0:5). Then (a) A is a (q,2 _q)-fuzzy bi-ideal of S. (b) A is an (2; 2 _q)-fuzzy bi-ideal of S. Proof. (a) Let x; y 2 S x y and t 2 (0; 1] be such that yt qA. Then y 2 B and so x 2 B. Thus if t 0:5 then A(x) 0:5 t: Hence xt 2 A: If t > 0:5; then A(x) + t > 0:5 + 0:5 = 1 and so xt qA: It follows that xt 2 _qA: Let x; y 2 S and r; t 2 (0; 1] be such that xr qA and yt qA. Then x; y 2 B and we have xy 2 B. If minfr; tg 0:5, then A(xy) 0:5 minfr; tg and hence (xy)m infr;tg 2 A. If minfr; tg > 0:5, then A(xy) + minfr; tg > 0:5 + 0:5 = 1 and so (xy)m infr;tg qA. Therefore (xy)m infr;tg 2 _qA. Now, let x; y; z 2 S and r; t 2 (0; 1] be such that xr qA and zt qA. Then x; z 2 B and we have xyz 2 B. If minfr; tg 0:5, then A(xyz) 0:5 minfr; tg and hence (xyz)m infr;tg 2 A. If minfr; tg > 0:5, then A(xyz) + minfr; tg > 0:5 + 0:5 = 1 and so (xyz)m infr;tg qA. Therefore (xyz)m infr;tg 2 _qA. Therefore A is a (q; 2 _q)-fuzzy bi-ideal of S. (b) Let x; y 2 S x y and t 2 (0; 1] be such that yt 2 A. Then A(y) t. Thus y 2 B and so x 2 B: If t 0:5 then A(x) 0:5 t: Hence xt 2 A: If t > 0:5; then A(x) + t > 0:5 + 0:5 = 1 and so xt qA: It follows that xr 2 _qA: Let x; y 2 S and r; t 2 (0; 1] be such that xr 2 A and yt 2 A. Then x; y 2 B and we have xy 2 B. If minfr; tg 0:5, then A(xy) 0:5 minfr; tg and hence (xy)m infr;tg 2 A. If minfr; tg > 0:5, then A(xy) + minfr; tg > 0:5 + 0:5 = 1 and so (xy)m infr;tg qA. Therefore (xy)m infr;tg 2 _qF . Now let x; y; z 2 S and and r; t 2 (0; 1] be such that xr 2 A and zt 2 A. Then x; z 2 B and we have xyz 2 B. If minfr; tg 0:5, then A(xy) 0:5 minfr; tg and hence (xyz)m infr;tg 2 A. If minfr; tg > 0:5, then A(xyz) + minfr; tg > 0:5 + 0:5 = 1 and so (xyz)m infr;tg qA. Therefore (xyz)m infr;tg 2 _qF and so A is an (2,2 _q)-fuzzy bi-ideal of S. From Example 8, we see that an (2; 2 _q)-fuzzy bi-ideal is not a (q,2 _q)fuzzy bi-ideal (Example 8, Part iv).
8
4
(2; 2 _q)-fuzzy bi-ideals
It is well known that the ideal theory plays a fundamental role in the developement of ordered semigroups. In [6], Jun et al. introduced the concept of a generalized fuzzy interior ideal of a semigroup. In this section we de…ne the notions of (2; 2 _q)-fuzzy bi-ideals of an ordered semigroup and investigate some of their properties in terms of (2; 2 _q)-fuzzy bi-ideals. Lemma 14 If =2 and =2 _q in de…nition 7. Then (B7 ), (B8 ) and (B9 ) respectively, of de…nition 7, are equivalent to the following conditions: (B10 ) (8x; y 2 S)(x y =) A(x) minfA(y); 0:5g): (B11 ) (8x; y 2 S)(A(xy) minfA(x); A(y); 0:5g). (B12 ) (8x; y; z 2 S)(A(xyz) minfA(x); A(z); 0:5g). Remark 15 A fuzzy subset A of an ordered semigroup S is an (2; 2 _q)-fuzzy bi-ideal of S if and only if it satis…es conditions (B10 ), (B11 ) and (B12 ) of the above Lemma. Remark 16 By the above Remark every fuzzy bi-ideal of an ordered semigroup S is an (2; 2 _q)-fuzzy bi-ideal of S. However, the converse is not true, in general. Example 17 Consider the ordered semigroup given in Example 8, and de…ne a fuzzy subset A : S ! [0; 1] by:
A(a) = 0:8;
A(b) = 0:7;
A(e) = 0:6;
A(d) = 0:5;
A(c) = 0:3:
Clearly A is an (2; 2 _q)-fuzzy bi-ideal of S. But A is not an ( ; )-fuzzy bi-ideal of S as shown in Example 8. Using Lemma 13, we have the following characterization of fuzzy bi-ideals of ordered semigroups. Proposition 18 Let (S; :; ) be an ordered semigroup and ; = 6 B S. Then B is a bi-ideal of S if and only if the characteristic function B of B is an (2; 2 _q)-fuzzy bi-ideal of S. In the following Theorem we give a condition for an (2; 2 _q)-fuzzy bi-ideal to ba an (2; 2)-fuzzy bi-ideal of S. Theorem 19 Let A be an (2; 2 _q)-fuzzy bi-ideal of S such that A(x) < 0:5 for all x 2 S. Then A is an (2; 2)-fuzzy bi-ideal of S. Proof. Let x; y 2 S; x By hypothesis, A(x)
y and t 2 (0; 1] be such that yt 2 A. Then A(y) minfA(y); 0:5g 9
minft; 0:5g = t;
t:
so x 2 At . Let x; y 2 S and t; r 2 (0; 1] be such that xt ; yr 2 A: Then A(x) t and A(y) r and so A(xy) minfA(x); A(y); 0:5g minft; r; 0:5g =minft; rg hence (xy)m inft;rg 2 A: Now, let x; y; z 2 S and t; r 2 (0; 1] be such that xt ; zr 2 A. Then A(x) t and A(z) r and we have A(xyz)
minfA(x); A(z); 0:5g
minft; r; 0:5g;
conseqently, (xyz)m inft;rg 2 A: Therefore A is an (2; 2)-fuzzy bi-ideal of S. For any fuzzy subset A of an ordered semigroup S and t 2 (0; 1], we denote Q(A; t) := fx 2 Sjxt qAg and [A]t := fx 2 Sjxt 2 _qAg: Obviously, [A]t = U (A; t) [ Q(A; t): We call [A]t an (2 _q)-level bi-ideal of A and Q(A; t) a q-level bi-ideal of A. We gave a characterization of (2; 2 _q)-fuzzy bi-ideals by using level subsets (see [8]). Now we provide another characterization of (2; 2 _q)-fuzzy bi-ideals by using the set [A]t : Theorem 20 Let S be an ordered semigroup and A a fuzzy subset of S. Then A is an (2; 2 _q)-fuzzy bi-ideal of S if and only if [A]t is a bi-ideal of S for all t 2 (0; 1]. Proof. =) : Let A be an (2; 2 _q)-fuzzy bi-ideal of S: Let x; y 2 S; x y and t 2 (0; 1] be such that y 2 [A]t . Then yt 2 _qA, that is, A(y) t or A(y) + t > 1. Since A is an (2; 2 _q)-fuzzy bi-ideal of S and x y; we have A(x) minfA(y); 0:5g: We have the following cases: Case 1 A(y) t. If t > 0:5; then A(x) minfA(y); 0:5g = 0:5 and so A(x) + t > 0:5 + 0:5 = 1: Hence xt qA. If t 0:5, then A(x) minfA(y); 0:5g Case 2 A(y) + t > 1. If t > 0:5, then A(x)
minfA(y); 0:5g
minf1
that is, A(x) + t > 1 and thus xt qA. If t A(x)
minfA(y); 0:5g
t, and hence xt 2 A.
t; 0:5g = 1
t;
0:5, then
minf1
t; 0:5g = 0:5
t
and so xt 2 A. Thus in both cases, we have xt 2 _qA and so x 2 [A]t . Let x; y 2 [A]t for t 2 (0; 1]: Then xt 2 _qA and yt 2 _qA, that is, A(x) t or A(x) + t > 1, and A(y) t or A(y) + t > 1. Since A is an (2; 2 _q)-fuzzy bi-ideal of S, we have, A(xy) Case 1 Let A(x)
minfA(x); A(y); 0:5g:
t and A(y) A(xy)
t. If t > 0:5. Then
minfA(x); A(y); 0:5g = 0:5 10
and hence (xy)t qA: If t
0:5: Then
A(xy)
minfA(x); A(y); 0:5g
t
and so (xy)t 2 A. Hence (xy)t 2 _qA: Case 2 Let A(x) t and A(y) + t > 1. If t > 0:5, then A(xy)
minfA(x); A(y); 0:5g:
If t > 0:5. Then A(xy)
minfA(x); A(y); 0:5g = minfA(y); 0:5g > minf1 t; 0:5g = 1 t;
i.e., A(xy) + t > 1 and thus (xy)t qA. If t A(xy)
minfA(x); A(y); 0:5g
0:5. Then minft; 1
t; 0:5g = t
and so (xy)t 2 A. Hence (xy)t 2 _qA. Case 3 Let A(x) + t > 1 and A(y) t. If t < 0:5, then A(xy)
minfA(x); A(y); 0:5g
minfA(x); 0:5g
minf1
t; 0:5g = 1
t;
i.e., A(xy) + t > 1 and hence (xy)t qA. If t < 0:5. Then A(xy)
minfA(x); A(y); 0:5g minf1 t; t; 0:5g = t
and so (xy)t 2 A. Hence (xy)t 2 _qA. Case 4 Let A(x) + t > 1 and A(y) + t > 1: If t > 0:5, then A(xy)
minfA(x); A(y); 0:5g > minf1
i.e., A(xy) + t > 1 and thus (xy)t qA. If t A(xy)
minfA(x); A(y); 0:5g
t; 0:5g = 1
t;
0:5: Then minf1
t; 0:5g = 0:5
t;
and so (xy)t 2 A. Thus in any case, we have (xy)t 2 _qA. Therefore xy 2 [A]t . Now, let x; z 2 [A]t for t 2 (0; 1] and y 2 S. Then xt 2 _qA and zt 2 _qA, that is, A(x) t or A(x) + t > 1, and A(z) t or A(z) + t > 1. Since A is an (2; 2 _q)-fuzzy bi-ideal of S, we have, A(xyz) Case 1 Let A(x)
minfA(x); A(z); 0:5g:
t and A(z) A(xyz)
t. If t > 0:5. Then
minfA(x); A(z); 0:5g = 0:5
11
and hence (xyz)t qA: If t
0:5: Then
A(xyz)
minfA(x); A(z); 0:5g
t
and so (xyz)t 2 A. Hence (xyz)t 2 _qA: Case 2 Let A(x) t and A(z) + t > 1. If t > 0:5. Then A(xyz)
minfA(x); A(z); 0:5g:
If t > 0:5. Then A(xyz)
minfA(x); A(z); 0:5g = minfA(z); 0:5g > minf1 t; 0:5g = 1 t;
i.e., A(xyz) + t > 1 and thus (xyz)t qA. If t A(xyz)
0:5. Then
minfA(x); A(z); 0:5g minft; 1 t; 0:5g = t
and so (xyz)t 2 A. Hence (xyz)t 2 _qA. Case 3 Let A(x) + t > 1 and A(z) t. If t < 0:5. Then A(xyz)
minfA(x); A(z); 0:5g minfA(x); 0:5g minf1 t; 0:5g = 1 t;
i.e., A(xyz) + t > 1 and hence (xyz)t qA. If t < 0:5. Then A(xyz)
minfA(x); A(z); 0:5g minf1 t; t; 0:5g = t
and so (xyz)t 2 A. Hence (xyz)t 2 _qA. Case 4 Let A(x) + t > 1 and A(z) + t > 1: If t > 0:5. Then A(xyz)
minfA(x); A(z); 0:5g > minf1 t; 0:5g = 1 t;
i.e., A(xyz) + t > 1 and thus (xyz)t qA. If t 0:5: Then A(xyz)
minfA(x); A(z); 0:5g
minf1
t; 0:5g = 0:5
t;
and so (xyz)t 2 A. Thus in any case, we have (xyz)t 2 _qA. Therefore xyz 2 [A]t . Conversely, let A be a fuzzy subset of S and t 2 (0; 1] be such that [A]t is a 12
bi-ideal of S. If possible, let A(x) < t 1, a contradiction. Hence A(x)
minfA(y); 0:5g for all x
y:
Let x; y 2 S be such that A(xy) < t 1, a contradiction. Hence A(xy) minfA(x); A(y); 0:5g for all x; y 2 S. Now let A(xay)
