SEMIGROUPS
Muhammad Shabir
Department of Mathematics, Quaid-i-Azam University, Islamabad-45320, Pakistan
[email protected] Muhammad Shoaib Arif
Department of Mathematics, Air University, Islamabad, Pakistan [email protected]
Asghar Khan
Department of Mathematics COMSATS Institute of Information Technology, Abbottabad, Pakistan
[email protected] Muhammad Aslam
Department of Mathematics, Quaid-i-Azam University, Islamabad-45320, Pakistan
Abstract: In this paper, we introduce the notions of intuitionistic fuzzy prime (resp. strongly prime and semiprime) bi-ideals of a semigroup. By using these ideas we characterize those semigroups for which each intuitionistic fuzzy bi-ideal is semiprime and strongly prime.
Key words: Intuitionistic fuzzy set; intuitionistic fuzzy bi-ideal; intuitionistic fuzzy prime (resp. strongly prime and semiprime) bi-ideal; regular and intra-regular semigroups.
1. Introduction
The theory of fuzzy sets proposed by Zadeh [30] in his classic paper of 1965, deal with the applications of fuzzy technology in information processing. The infor- mation processing is already important and it will certainly increase in importance in the future. Mordeson, Malik and Kuroki gave a systematic exposition of fuzzy semigroups in [23], where one can …nd theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy …nite state machines and fuzzy languages. The
1
monograph by Mordeson and Malik [24] deals with the applications of fuzzy ap- proach to the concepts of automata and formal languages. After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fun- damental concept. Atanassov [5] introduced the notion of intuitionistic fuzzy sets which is a generalization of fuzzy sets. The fuzzy sets give the degree of membership of an element in a given set while the intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership. As for fuzzy sets, the degree of membership is a real number between 0 and 1, this is also the case for the degree of non-membership, and the sum of these two degrees is not greater than 1. For more details on intuitionistic fuzzy sets, we refer the reader to [5,6,7]. Fuzzy sets are intuitionistic fuzzy sets but the converse is not necessarily true [5]. Infect, there are situations where intuitionistic fuzzy set theory is more appropriate to deal with [9]. Intuitionistic fuzzy set theory has been applied in di¤erent …elds, that is, logic programming, decision making problems, etc. De et al. [12] studied the Sanchez’s approach for medical diagnosis and extended this concept with the notion of intu- itionistic fuzzy set theory. Some authors applied this concept to generalize some notions of algebra, for example Davvaz et al. [11], applied this concept in Hv- modules. They introduced the notion of an intuitionistic fuzzy Hv-submodule of anHv-module, and studied related properties. Kim et al. [20], considered the intu- itionistic fuzzi…cation of the concept of sub-hyperquasigroups in a hyperquasigroup.
In [21], [22], Kim and Jun introduced the concept of intuitionistic fuzzy (interior) ideals of semigroups.
Shabir et al. in [26], introduced the concept of prime bi-ideals, strongly prime bi-ideal and semiprime bi-ideals of a semigroup and studied those semigroups for which each bi-ideal is semiprime and strongly irreducible.
In this paper we introduce the concept of intuitionistic fuzzy prime, strongly prime and semiprime bi-ideals of semigroups and give characterizations of semi- groups in terms of these notions. We characterize those semigroups for which each intuitionistic fuzzy bi-ideal is semiprime and strongly irreducible.
2. Basic Concepts of Semigroups
A semigroup is a non-empty setStogether with an associative binary operation
“ ”. An element 0 of a semigroup S with at least two elements is called a zero element of S if x0 = 0x= 0 for allx in S. A semigroup which contains a zero element is called a semigroup with zero. If a semigroupS has no zero element then it is easy to adjoin a zero element0 to the set by de…ning0x=x0 = 0 = 00for all xinS. We shall use the notationS0with the following meanings:
S0= S, if S has a zero element S[ f0g, otherwise.
ForA; B S, we denote,
AB:=fabja2A; b2Bg.
A non-empty subsetAof a semigroupSis called a subsemigroup ofSifab2A for all a; b 2 A. A subsemigroup B of a semigroup S is called a bi-ideal of S if BSB B. A non-empty subsetAof a semigroupS is called left (right) ideal ofS ifSA A(AS A). Ais called two sided ideal ofSif it is both a left and a right ideal of S. Every left (right) ideal of a semigroup is a bi-ideal but the converse is not true. It is well known that the intersection of any number of bi-ideals of a
semigroupS is either empty or a bi-ideal of S. Also the product of two bi-ideals of a semigroup S is a bi-ideal of S. Let a 2S. Then intersection of all bi-ideals ofS which containa is a bi-ideal ofS containinga. Of course this is the smallest bi-ideal ofScontainingaand is called the bi-ideal generated bya. We shall denote this bi-ideal byB(a). ClearlyB(a) =a[a2[aSa.
A bi-ideal B of a semigroup S is called prime (strongly prime) if B1B2 B (B1B2\B2B1 B) impliesB1 B or B2 B for any bi-ideals B1 and B2 of S (cf.[26]). A bi-idealB of a semigroupS is called semiprime if B12 B implies B1 B for any bi-idealB1ofS (cf.[26]).
An element a of a semigroup S is called a regular element if there exists an elementxinSsuch thataxa=a. A semigroupS is called regular if every element ofS is regular. An elementaof a semigroupS is called intra-regular if there exist elementsxandy inS such thatxa2y=a. A semigroupS is called intra-regular if every element ofS is intra regular.
A function f from a non-empty set A to the unit interval [0;1] of real num- bers is called a fuzzy subset of A. A fuzzy subset f of a semigroup S is called a fuzzy subsemigroup of S if f(xy) minff(x); f(y)g for all x; y2 S. A fuzzy subsemigroup ofSis called a fuzzy bi-ideal ofSiff(xyz) minff(x); f(z)gfor all x; y; z2S(cf.[23]). A fuzzy subsetf of a semigroupS is called a fuzzy left (right) ideal ofS iff(xy) f(y) (f(xy) f(x)) for all x; y2S. Every fuzzy left (right) ideal is a fuzzy bi-ideal but the converse is not true. For any two fuzzy subsetsf; g of a non-empty setA we de…ne the fuzzy subsets
(f^g)(a) =f(a)^g(a)
(f_g) (a) =f(a)_g(a) for alla2A.
Alsof g means thatf(a) g(a)for alla2A.
3. Intuitionistic fuzzy bi-ideals
As an important generalization of the notion of fuzzy set inS, Atanassov (cf.[5]) introduced the concept of an intuitionistic fuzzy set (IF S for short) de…ned on a non-empty setS as objects having the form
A=fhx; A(x); A(x)ijx2Sg,
where the functions A : S ! [0;1]and A : S ! [0;1] denote the degree of membership(namely A(x)) and thedegree of non-membership (namely A(x)) of each elementx2S to the set A respectively, and 0 A(x) + A(x) 1 for all x2S (cf.[5]). In particular, 0 and 1 denote the intuitionistic fuzzy empty set and the intuitionistic fuzzy whole set in a set X de…ned by 0 (x) = (0;1) and 1 (x) = (1;0)for each x2X, respectively. We shall write IF S for intuitionistic fuzzy sets.
For simplicity we useA= ( A; A)forIF Sinstead ofA=fhx; A(x); A(x)ijx2 Sg.
For any twoIFSsA= ( A; A)andB= ( B; B)of a semigroupS we de…ne:
(1)A B if and only if A(x) B(x)and A(x) B(x)for allx2S, (2)Ac=fhx; A(x); A(x)ijx2Sg= ( A; A),
(3)A\B=fhx;minf A(x); B(x)g;maxf A(x); B(x)gijx2Sg
= ( A^ B; A_ B),
(4)A[B=fhx;maxf A(x); B(x)g;minf A(x); B(x)gijx2Sg
= ( A_ B; A^ B),
(5)1 = (1;0) and0 = (0;1).
For any two intuitionistic fuzzy sets A = ( A; A) and B = ( B; B) of a semigroupS, de…neA B= ( A B; A B)(cf. [19]) where:
A B(a) :=
( W
a=yz
minf A(y); B(z)gifa=yz,
0 otherwise
and
A B(a) :=
( V
a=yz
maxf A(y); B(z)if a=yz,
0 otherwise.
IfIF S(S)denotes the set of all intuitionistic fuzzy sets of a semigroupS. One can easily see that the multiplication “ ” onIF S(S)is associative.
It is clear that for anyIF S; A; B; Cof a semigroupS ifB C, then A B A C andB A C A: (cf:[1])
3.1. Lemma [1]. LetSbe a semigroup andA; B; Cbe any intuitionistic fuzzy subsets ofS then,
(1)A (B[C) = (A B)[(A C),(B[C) A= (B A)[(C A).
(2)A (B\C) (A B)\(A C),(B\C) A (B A)\(C A).
Let fAigi2I be a family of intuitionistic fuzzy subsets of S then T
i2I
Ai is an intuitionistic fuzzy subset ofS where T
i2I
Ai= (V
i2I Ai;W
i2I Ai)and V
i2I Ai : S ![0;1]jx ! V
i2I Ai(x) :=infi2If Ai(x)jx2Sg, W
i2I Ai : S ![0;1]jx ! W
i2I Ai(x) :=supi2If Ai(x)jx2Sg.
LetAbe a non-empty subset of a semigroupS, the intuitionistic characteristic function A= (
A;
A)is de…ned as [19]:
A(x) := 1ifx2A, 0ifx =2A.
and
A(x) := 0 ifx2A, 1 ifx =2A.
3.2. De…nition [16]. Let S be a semigroup and let A = ( A; A) be an intuitionistic fuzzy subsets ofS thenA is called an:
(1) Intuitionistic fuzzy subsemigroup ofS, if
A(xy) minf A(x); A(y)g
and A(xy) maxf A(x); A(y)gfor allx; y2S.
(2) Intuitionistic fuzzy bi-ideal ofS, ifAis an intuitionistic fuzzy subsemigroup ofS and
A(xyz) minf A(x); A(z)g
and A(xyz) maxf A(x); A(z)g for allx; y; z2S.
(3) Intuitionistic fuzzy left (right) ideal ofS, if
A(xy) A(y) ( A(xy) A(x))
and A(xy) A(y) ( A(xy) A(x)) for allx; y; z2S.
Every intuitionistic fuzzy left (right) ideal is intuitionistic fuzzy bi-ideal but the converse is not true.
3.3. Lemma [20]. LetAbe a non-empty subset of a semigroupS. Then (1)Ais a subsemigroup ofS if and only if A= (
A;
A)is an intuitionistic fuzzy subsemigroup ofS.
(2)Ais a bi-ideal ofS if and only if A= (
A;
A)is an intuitionistic fuzzy bi-ideal ofS.
3.4. Lemma [1]. LetS be a semigroup and A; B S then (i) A B if and only if A B.
(ii) A B = A B.
3.5. Lemma [1]. LetSbe a semigroup and0 6=Abe an intuitionistic fuzzy subset ofS. Then
(i) Ais an intuitionistic fuzzy subsemigroup ofS if and only ifA A A.
(ii) A is an intuitionistic fuzzy bi-ideal of S if and only if A A A and A 1 A A.
3.6. Lemma [1]. Every intuitionistic fuzzy left (right) ideal of a semigroup is intuitionistic fuzzy bi-ideal.
3.7. Lemma [1]. A semigroupS is regular if and only ifA B =A\B for each intuitionistic fuzzy right ideal A and each intuitionistic fuzzy left idealB of S.
3.8. Lemma [1]. A semigroupS is intra regular if and only ifA\B B A for each intuitionistic fuzzy right ideal Aand each intuitionistic fuzzy left idealB ofS.
4. Intuitionistic fuzzy Prime (strongly prime and semiprime) bi-ideals In this section we de…ne intuitionistic fuzzy prime (resp. strongly prime and semiprime) bi-ideals of a semigroupS.
4.1. De…nition. Let S be a semigroup and A = ( A; A) an intuitionistic fuzzy bi-ideal ofS. ThenAis called an intuitionistic fuzzy prime (strongly prime) bi-ideal ofS if:
For any intuitionistic fuzzy bi-ideals B = ( B; B) and C = ( c; c) of S, B C A(resp. B C\C B A) impliesB AorC A.
An intuitionistic fuzzy bi-ideal A = ( A; A) of S is called an intuitionistic fuzzy semiprime bi-ideal ofS if:
For any intuitionistic fuzzy bi-idealB = ( B; B)ofS,B B AimpliesB A. Obviously every intuitionistic fuzzy strongly prime bi-ideal is an intuitionistic fuzzy prime bi-ideal and every intuitionistic fuzzy prime bi-ideal is semiprime but the converse is not true is general.
4.2. Theorem. A bi-ideal B of a semigroup S is prime if and only if the intuitionistic characteristic function B = (
B;
B)ofB is an intuitionistic fuzzy prime bi-ideal ofS.
Proof. Suppose B is a prime bi-ideal of S, then by Lemma 3.3, B is an intuitionistic fuzzy bi-ideal ofS. LetA,Cbe any intuitionistic fuzzy bi-ideals ofS such thatA C BbutA* B andC* B, then there existx; y2Ssuch that
A(x)6= 0; A(x)6= 1and c(y)6= 0; c(y)6= 1 but
B(x) = 0;
B(x) = 1 and
B(y) = 0;
B(y) = 1.
Hence x =2 B and y =2 B. Since B is a prime bi-ideal of S, therefore we have B(x)B(y)*B.
Since
A(x)6= 0; A(x)6= 1and C(y)6= 0; C(y)6= 1, therefore
minf A(x); C(y)g 6= 0and max{ A(x); C(y)g 6= 1.
Since B(x)B(y) * B, therefore there exists a 2 S such that a 2 B(x)B(y) but a =2B. Thus we have
B(a) = 0; B(a) = 1 and hence
A C(a) = 0and A C(a) = 1:
Sincea2B(x)B(y)we havea=x1y1for some x12B(x)andy12B(y). Thus
A C(a) = W
a=x1y1
minf A(x1); c(y1)g minf A(x1); c(y1)g and
A C(a) = V
a=x1y1
maxf A(x1); c(y1)g maxf A(x1); c(y1)g.
Since,x12B(x) =fxg [ fx2g [xSx, we havex1=xorx1=x2 orx1=xyxfor somey 2S.
Ifx1=xthen
A(x1) = A(x), and A(x1) = A(x).
Ifx1=x2then
A(x1) = A(x2) minf A(x); A(x)g= A(x) and
A(x1) = A(x2) maxf A(x); A(x)g= A(x).
Ifx1=xyxthen
A(x1) = A(xyx) minf A(x); A(x)g= A(x) and
A(x1) = A(xyx) maxf A(x); A(x)g= A(x).
Also sincey1 2B(y) =fyg [ fy2g [ySy; we havey1 =y ory1=y2 ory1=yxy for somex2S:Ify1=y then
C(y1) = C(y);and C(y1) = C(y):
Ify1=y2then
C(y1) = C(y2) minf C(y); C(y)g
= C(y) and
C(y1) = C(y2) maxf C(y); C(y)g= C(y):
Ify1=yxythen
C(y1) = C(yxy) minf C(y); C(y)g
= C(y) and
C(y1) = C(yxy) maxf C(y); C(y)g= C(y):
Thus
A C(a) minf A(x1); c(y1)g minf A(x); c(y)g 6= 0 and
A C(a) maxf A(x1); c(y1)g maxf A(x); c(y)g 6= 1
which is a contradiction to the fact that A C(a) = 0and A C(a) = 1.
Thus for any intuitionistic fuzzy bi-idealsA; C ofS, A C B =)A B orC B.
Conversely, assume that B is an intuitionistic fuzzy prime bi-ideal ofS. Let B1; B2be any bi-ideals ofS such thatB1B2 B, then by Lemma 3.3, B1and B2 are intuitionistic fuzzy bi-ideals ofS. By hypothesis, B1 B2 B. By Lemma 3.4,
B1 B2 = B1 B2. Thus B1 B2 B. Since B is prime, we have B1 B or B2 B. By Lemma 3.4, we haveB1 B orB2 B.
Similarly we can prove the following:
4.3. Theorem. A bi-ideal B of a semigroup S is semiprime if and only if the intuitionistic characteristic function B of B is intuitionistic fuzzy semiprime bi-ideal ofS.
4.4. Theorem. A bi-idealB of a semigroupSis strongly prime if and only if the intuitionistic characteristic function B = (
B;
B)of B is an intuitionistic fuzzy strongly prime bi-ideal ofS.
Proof. SupposeBis a strongly prime bi-ideal ofS then by Lemma 3.3, B is an intuitionistic fuzzy bi-ideal of S. LetA, C be any intuitionistic fuzzy bi-ideals of S such that A C\C A B but A * B and C * B, then there exist x; y2S such that
A(x)6= 0; A(x)6= 1 but
B(x) = 0;
B(x) = 1 and
B(y) = 0;
B(y) = 1.
Hence x =2 B and y =2 B. Since B is a strongly prime bi-ideal of S, we have B(x)B(y)\B(y)B(x)*B.
Since
A(x)6= 0; A(x)6= 1and C(y)6= 0; C(y)6= 1 we have
minf A(x); C(y)g 6= 0and max{ A(x); C(y)g 6= 1.
Since B(x)B(y)\B(y)B(x)* B; so there existsa 2 S such that a2 B(x)B(y) anda2B(y)B(x), buta =2B. Thus we have
B(a) = 0;
B(a) = 1 and hence
A C(a)^ C A(a) = 0and A C(a)_ C A(a) = 1.
Sincea2B(x)B(y)and a2B(y)B(x), we have a=x1y1 anda=y2x2 for some x1; x22B(x)andy1; y22B(y). Thus
A C(a) = W
a=x1y1
minf A(x1); c(y1)g minf A(x1); c(y1)g and
A C(a) = V
a=x1y1
maxf A(x1); c(y1)g maxf A(x1); c(y1)g.
Since,x12B(x) =fxg [ fx2g [xSx;we havex1=xorx1=x2 orx1=xyxfor somey 2S.
Ifx1=xthen
A(x1) = A(x); and A(x1) = A(x):
Ifx1=x2then
A(x1) = A(x2) minf A(x); A(x)g= A(x) and
A(x1) = A(x2) maxf A(x); A(x)g= A(x).
Ifx1=xyxthen
A(x1) = A(xyx) minf A(x); A(x)g= A(x) and
A(x1) = A(xyx) maxf A(x); A(x)g= A(x).
Also sincey1 2B(y) =fyg [ fy2g [ySy; we havey1 =y ory1=y2 ory1=yxy for somex2S:Ify1=y then
C(y1) = C(y); and C(y1) = C(y).
Ify1=y2then
C(y1) = C(y2) minf C(y); C(y)g= C(y) and
C(y1) = C(y2) maxf C(y); C(y)g= C(y).
Ify1=yxythen
C(y1) = C(yxy) minf C(y); C(y)g= C(y)
and
C(y1) = C(yxy) maxf C(y); C(y)g= C(y).
Thus
A C(a) minf A(x1); c(y1)g minf A(x); c(y)g 6= 0 and
A C(a) maxf A(x1); c(y1)g maxf A(x); c(y)g 6= 1
Similarly we can show that C A(a)>0 and C A(a)6= 1. Thus, A C(a)^
C A(a)>0and A C(a)_ C A(a)6= 1which is a contradiction to the fact that
A C(a)^ C A(a) = 0 and A C(a)_ C A(a) = 1. Thus, for any intuitionistic fuzzy bi-idealsA; C ofS,A C\C A B=)A B or C B.
Conversely, assume that B is an intuitionistic fuzzy strongly prime bi-ideal of S. LetB1; B2 be bi-ideals ofS such that B1B2\B2B1 B, then by Lemma 3.3,
B1and B2are intuitionistic fuzzy bi-ideals ofS. By hypothesis, B1 B2\ B2 B1
B. By Lemma 3.4, we have B1 B2 = B1 B2. Thus, B1 B2\ B2 B1 B. Since B is strongly prime, we have B1 B or B2 B. By Lemma 3.4, we haveB1 B or B2 B.
4.5. De…nition. An intuitionistic fuzzy bi-idealB ofS is called idempotent if
B =B2=B B, that is, B B= B B = B; B B= B B = B. 4.6. Lemma. [21]. LetfAigi2I be a family of intuitionistic fuzzy bi-ideals of S, then T
i2I
Ai is an intuitionistic fuzzy bi-ideal ofS.
4.7. De…nition. Let S be a semigroup and A = ( A; A) an intuitionistic fuzzy bi-ideal of S. Then A is called an intuitionistic fuzzy irreducible (resp.
strongly irreducible) bi-ideal ofS if:
For any intuitionistic fuzzy bi-ideals B = ( B; B) and C = ( c; c) of S, B\C=A (resp. B\C A) impliesB =Aor C=A (resp. B A orC A).
The proof of the following Lemma is straight forward, so we omit it.
4.8. Lemma. A bi-idealB of a semigroupS is an irreducible (resp. strongly irreducible) if and only if the intuitionistic characteristic function B= (
B;
B) ofB is an intuitionistic fuzzy irreducible (resp. strongly irreducible) bi-ideal ofS.
5. Semigroup in which each intuitionistic fuzzy bi-ideal is strongly prime
In this section we study those semigroups in which each intuitionistic fuzzy bi-ideal is semiprime and also those semigroups in which each intuitionistic fuzzy bi-ideal is strongly prime.
5.1. Lemma. LetS be a semigroup. Then the intersection of any family of intuitionistic fuzzy prime bi-ideals ofS is an intuitionistic fuzzy semiprime bi-ideal ofS.
Proof. Straight forward.
5.2. Proposition. An intuitionistic fuzzy strongly irreducible, semiprime bi- ideal of a semigroupS is an intuitionistic fuzzy strongly prime bi-ideal ofS.
Proof. LetAbe an intuitionistic fuzzy strongly irreducible, semiprime bi-ideal ofS. LetBandCbe intuitionistic fuzzy bi-ideals ofSsuch thatB C\C B A.
Since(B\C)2 B C and also(B\C)2 C B, so(B\C)2 B C\C B A.
Since A is an intuitionistic fuzzy semiprime bi-ideal, so (B\C) A. As A is irreducible soB Aor C A, that isA is an intuitionistic fuzzy strongly prime bi-ideal.
5.3. Proposition. Let A = ( A; A) be an intuitionistic fuzzy bi-ideal of a semigroupS with A(a) =t and A(a) = 1 t wherea2S and t 2(0;1], then there exist an intuitionistic fuzzy irreducible bi-idealB= ( B; B)ofS such that A B and B(a) =t; B(a) = 1 t.
Proof. Let X = fCjC is an intuitionistic fuzzy bi-ideal of S, C(a) = t;
C(a) = 1 t and A Cg; then X 6= because A 2X. Then the collectionX is a partially ordered set under inclusion. If Y is any totally order subset of X, sayY =fCiji2Ig, then S
i2I
Ci is an intuitionistic fuzzy bi-ideal ofS. Indeed: Let x; y2S, then
(W
i2I Ci)(xy) = W
i2I
( Ci(xy)) W
i2I
( Ci(x)^ Ci(y))
= W
i2I
( Ci(x))^ W
i2I
( Ci(y))
= (W
i2I Ci)(x)^(W
i2I Ci)(y) and
(V
i2I Ci)(xy) = V
i2I
( Ci(xy)) V
i2I
( Ci(x)_ Ci(y))
= V
i2I
( Ci(x))_ V
i2I
( Ci(y))
= (V
i2I Ci)(x)_(V
i2I Ci)(y) Hence S
i2I
Ci is an intuitionistic fuzzy subsemigroup ofS. Letx; y; z2S, then
(W
i2I Ci)(xyz) = W
i2I
( Ci(xyz)) W
i2I
( Ci(x)^ Ci(z))
= W
i2I
( Ci(x))^ W
i2I
( Ci(z))
= (W
i2I Ci)(x)^(W
i2I Ci)(z)
and
(V
i2I Ci)(xyz) = V
i2I
( Ci(xyz)) V
i2I
( Ci(x)_ Ci(z))
= V
i2I
( Ci(x))_ V
i2I
( Ci(z))
= (V
i2I Ci)(x)_(V
i2I Ci)(z).
Thus, S
i2I
Ciis an intuitionistic fuzzy bi-ideal ofS. AsA Cifor eachi2I, so
A S
i2I
Ci. Also,(W
i2I Ci)(a) = W
i2I Ci(a) =tand(V
i2I Ci)(a) = V
i2I Ci(a) = 1 t.
Thus, S
i2I
Ci is the upper bound of Y. Thus, by Zorn’s lemma, their exists an intuitionistic fuzzy bi-ideal B = ( B; B) of S which is maximal with respect to the property thatA B and B(a) =t; B(a) = 1 t. We now show that B is an intuitionistic fuzzy irreducible bi-ideal of S. Suppose B =B1\B2 where B1 andB2 are intuitionistic fuzzy bi-ideal ofS. ThenB B1 andB B2. We claim thatB=B1 orB=B2. Suppose on the contraryB6=B1andB6=B2. SinceB is a maximal bi-ideal with respect to the property that B(a) =tand B(a) = 1 t andB6=B1 andB6=B2, it follows that
B1(a)6=t or B1(a)6= 1 t and
B2(a)6=t or B2(a)6= 1 t:
Hence
t = B(a) = B1^B2(a) = B1(a)^ B2(a)6=t or1 t = B(a) = B1_B2(a) = B1(a)_ B2(a)6= 1 t;
which is a contradiction. Hence eitherB =B1orB =B2. Thus,Bis an irreducible intuitionistic fuzzy bi-ideal ofS.
5.4. Theorem. [1]. A semigroup S is regular if and only if for every intu- itionistic fuzzy bi-idealAofS we have,
A=A 1 A.
5.5. Lemma. LetSbe a semigroup andA,Bbe intuitionistic fuzzy bi-ideals ofS. ThenA B is an intuitionistic fuzzy bi-ideal ofS.
Proof. LetAandB be intuitionistic fuzzy bi-ideals ofS. Then (A B) (A B) = (A B A) B
(A 1 A) B
A B.
Also,
(A B) 1 (A B) (A 1 ) 1 (A B)
= A (1 1 ) (A B) A 1 (A B)
= (A 1 A) B
A B.
ThusA B is an intuitionistic fuzzy bi-ideal ofS.
Next Theorem characterizes those semigroups in which each intuitionistic fuzzy bi-ideal is semiprime.
5.6. Theorem. LetS be a semigroup. Then the following are equivalent:
(1)S is both regular and intra-regular.
(2)A A=A for every intuitionistic fuzzy bi-idealAofS.
(3)A\B = (A B)\(B A)for every intuitionistic fuzzy bi-idealsA andB ofS.
(4)Each intuitionistic fuzzy bi-ideal ofS is intuitionistic fuzzy semiprime.
(5)Each proper intuitionistic fuzzy bi-ideal ofSis the intersection of irreducible intuitionistic fuzzy semiprime bi-ideals ofS which contain it.
Proof. (1) =) (2) Let S be both regular and intra-regular semigroup and A= ( A; B)be an intuitionistic fuzzy bi-ideal ofS. Leta2S. SinceS is regular and intra-regular, so there existx; y; z2S such thata=axa anda=ya2z. Thus we havea=axa=axaxa=ax(ya2z)xa= (axya)(azxa). Hence,
A A(a) = W
a=yz
minf A(y); A(z)g minf A(axya); A(azxa)g
minf( A(a)^ A(a)); A(a)^ A(a)g
= A(a).
and
A A(a) = V max
a=yz f A(y); A(z)g maxf A(axya); A(azxa)g
maxf( A(a)_ A(a));( A(a)_ A(a))g
= A(a).
Hence A A(a) A(a)and A A(a) A(a). ThusA A A. For the reverse inclusion, sinceAis an intuitionistic fuzzy subsemigroup ofS, we have A A A and A A A, that isA A A. Therefore,A A=A.
(2) =) (3): Let A; B be intuitionistic fuzzy bi-ideals of S, then A\B is an intuitionistic fuzzy bi-ideal ofS:By (2), we have
A\B= (A\B) (A\B) A B;
similarly,A\B B A:Thus
A\B A B\B A.
For the reverse inclusion, sinceA BandB Aare intuitionistic fuzzy bi-ideals ofS, soA B\B Ais an intuitionistic fuzzy bi-idealS. Hence by hypothesis, we have
A B\B A = (A B\B A) (A B\B A) A B B A=A (B B) A
= A B A(becauseB B =B )
A 1 A
= A (by Lemma 3.5).
Hence A B\B A A. Similarly, we can prove thatA B\B A B.
HenceA B\B A A\B. Therefore,A B\B A=A\B.
(3) =) (4). Let A and B be intuitionistic fuzzy bi-ideals of S such that A A B. Then by hypothesis,
A = A\A
= A A\A A
= A A.
ThusA B. Hence each intuitionistic fuzzy bi-ideal ofS is semiprime.
(4) =)(5). LetAbe a proper intuitionistic fuzzy bi-ideal ofSandfAi:i2Ig be the collection of all irreducible intuitionistic fuzzy bi-ideals of S which contain A. Proposition 5.3 guarantees the existence of such irreducible intuitionistic fuzzy bi-ideals. Hence A T
i2IAi. Let a2S, then by Proposition 5.3, their exist an irreducible intuitionistic fuzzy bi-ideal A of S such that A A and A(a) =
A (a); A(a) A (a). Thus A 2 fAi:i2Ig. Hence T
i2I
Ai A , that is V
i2I Ai(a) A(a) = A(a) and W
i2I Ai(a) A(a) A(a). Hence T
i2I
Ai A.
Consequently, T
i2I
Ai =A. By hypothesis, each intuitionistic fuzzy bi-ideal of S is semiprime, so each intuitionistic fuzzy bi-ideal ofSis the intersection of irreducible semiprime intuitionistic fuzzy bi-ideals ofS which contain it.
(5) =)(2): LetA be an intuitionistic fuzzy bi-ideal ofS. Then A A is also an intuitionistic fuzzy bi-ideal of S by Lemma 5.5. ThenA A A, becauseAis an intuitionistic fuzzy subsemigroup ofS. AlsoA A A A, by hypothesisA A is semiprime soA A A. ThusA A=A.
(2))(1): LetA be an intuitionistic fuzzy right ideal andB an intuitionistic fuzzy left ideal ofS,A B A\B. On the other hand,A\B is an intuitionistic fuzzy bi-ideal of S, so by hypothesis A\B = (A\B) (A\B) A B. Thus A\B =A B, which implies thatSis regular. AlsoA\B= (A\B) (A\B) B A which implies thatS is intra regular.
5.7. Proposition. LetS be a regular and intra regular semigroup. Then the following assertions for an intuitionistic fuzzy bi-idealAofS are equivalent.
(1)Ais strongly irreducible.
(2)Ais strongly prime.
Proof. (1) =)(2):LetSbe a regular and intra regular semigroup andAbe a strongly irreducible intuitionistic fuzzy bi-ideal ofS. SupposeB; Care intuitionistic fuzzy bi-ideal ofS such that (B C)\(C B) A. SinceS is both regular and
intra regular so by Theorem 5.6, (B C)\(C B) =B\C. Hence B\C A.
SinceAis strongly irreducible, we have B AorC A.
(2) =) (1): Suppose A is strongly prime intuitionistic fuzzy bi-ideal of S.
Let B, C be any intuitionistic fuzzy bi-ideals of S such that B\C A. As B C\C B=B\C A andAis strongly prime, so we have B AorC A.
ThusAis strongly irreducible intuitionistic fuzzy bi-ideal ofS.
5.8. Theorem. Each intuitionistic fuzzy bi-ideal of a semigroupS is strongly prime if and only ifS is regular and intra regular and the set of intuitionistic fuzzy bi-ideals ofS is totally ordered, by inclusion.
Proof. Suppose that each intuitionistic fuzzy bi-ideal of a semigroup S is strongly prime, then each intuitionistic fuzzy bi-ideal of the semigroup S is semi- prime, then by Theorem 5.6,Sis both regular and intra regular. We show that the set of intuitionistic fuzzy bi-ideal ofS is totally ordered by inclusion. LetB; C be intuitionistic fuzzy bi-ideal ofS, then by Theorem 5.6,(B C)\(C B) =B\C.
As each intuitionistic fuzzy bi-ideal of S is strongly prime so B \C is strongly prime, hence eitherB B\C orC B\C, that is eitherB C orC B.
Conversely, assume thatS is regular and intra-regular and the set of intuition- istic fuzzy bi-ideals ofSis totally ordered under inclusion, then we show that each intuitionistic fuzzy bi-ideal of the semigroupS is strongly prime. LetA be an in- tuitionistic fuzzy bi-ideal ofS andB; C any intuitionistic fuzzy bi-ideals ofS such that(B C)\(C B) A. SinceSis both regular and intra-regular, so by Theorem 5.6, (B C)\(C B) =B\C. HenceB\C A. Since the set of intuitionistic fuzzy bi-ideals of S is totally ordered, so either B C or C B, that is either B\C=B or B\C=C. Thus eitherB AorC A.
5.9. Theorem. If the set of intuitionistic fuzzy bi-ideals of a semigroupS is totally order by inclusion, then S is both regular and intra regular if and only if each intuitionistic fuzzy bi-ideal ofS is intuitionistic fuzzy prime bi-ideal of S.
Proof. SupposeS is both regular and intra regular and A any intuitionistic fuzzy bi-ideal ofS. LetB; Cbe intuitionistic fuzzy bi-ideals ofSsuch thatB C A. Since the set of intuitionistic fuzzy bi-ideal of S is totally ordered, therefore either B C or C B. Suppose that B C then B B B C A. By Theorem 5.6, A is semiprime, so B A. HenceA is a prime intuitionistic fuzzy bi-ideal ofS.
Conversely, assume that every intuitionistic fuzzy bi-ideal ofS is prime. Since every prime intuitionistic fuzzy bi-ideal is semiprime, so by Theorem 5.6,S is both regular and intra-regular.
5.10. Theorem. For a semigroupS,the following assertion are equivalent.
(1)Set of intuitionistic fuzzy bi-ideals ofS is totally ordered by inclusion . (2)Each intuitionistic fuzzy bi-ideal ofS is strongly irreducible.
(3)Each intuitionistic fuzzy bi-ideal ofS is irreducible.
Proof. (1) =) (2): Let A be an arbitrary intuitionistic fuzzy bi-ideal of S.
LetB; C be intuitionistic fuzzy bi-ideals ofS such thatB\C A. Since the set of intuitionistic fuzzy bi-ideals ofSis totally ordered by inclusion, so eitherB C orC B. Thus eitherB Aor C A. Hence Ais strongly irreducible.
(2) =)(3): LetAbe an arbitrary intuitionistic fuzzy bi-ideal ofS. Let B; C be intuitionistic fuzzy bi-ideals ofSsuch thatB\C=A. ThenA B andA C.
By hypothesis either B A orC A. Thus eitherB=AorC=A.
(3) =)(1): Suppose each intuitionistic fuzzy bi-ideal of S is irreducible. Let A; B be intuitionistic fuzzy bi-ideals of S, then A\B is an intuitionistic fuzzy bi-ideal ofS. AlsoA\B =A\B. By hypothesis eitherA=A\B orB=A\B, that is eitherA B or B A. Hence the set of intuitionistic fuzzy bi-ideals ofS is totally ordered by inclusion .
The next example shows that if each bi-ideal of a semigroup is prime, then it is not necessary that its each intuitionistic fuzzy bi-ideal is prime.
5.11. Example. Consider the semigroupS=f0; a; bg 0 a b
0 0 0 0 a 0 a a b 0 b b
It is evident thatS is both regular and intra regular.
Bi-ideals inS aref0g;f0; ag;f0; bg andS.
All bi-ideals are prime and hence semiprime bi-ideals.
Fact 1. Each intuitionistic fuzzy subset of S is an intuitionistic fuzzy sub- semigroup ofS.
Fact 2. Each intuitionistic fuzzy subsetA= ( A; A)ofSis an intuitionistic fuzzy bi-ideal ofS if and only if A(0) A(x)and A(0) A(x) for allx2S.
Proof. SupposeA= ( A; A)is an intuitionistic fuzzy bi-ideal ofS, then by de…nition
A(0) = A(x0x) A(x)^ A(x) = A(x)and
A(0) = A(x0x) A(x)_ A(x) = A(x)for allx2S.
Conversely, assume that
A(0) A(x)and A(0) A(x) for allx2S.
Thenxyz=xifx; y; z2 fa; bg andxyz= 0if one of them is zero. Thus
A(xyz) A(x)^ A(z)and A(xyz) A(x)_ A(z).
Fact 3. Since S is regular and intra regular, so every intuitionistic fuzzy bi-ideal ofS is intuitionistic fuzzy semiprime bi-ideal.
Consider the intuitionistic fuzzy bi-idealsA; B; C ofS,
A(0) = 0:7, A(a) = 0:6, A(b) = 0:4 and A(0) = 0:3, A(a) = 0:4,
A(b) = 0:4
B(0) = 1:0, B(a) = 0:5, B(b) = 0:4 and B(0) = 0, B(a) = 0:5,
B(b) = 0:4
C(0) = 0:7, C(a) = 0:65, C(b) = 0:3 and C(0) = 0:3, C(a) = 0:35,
C(b) = 0:7
B C(0) = W
0=xyf B(x)^ C(y)g= W
0=xyf1^0:7;1^0:65;1^0:3;0:5^0:7;0:4^0:7g= 0:7
B C(a) = W
a=xyf B(x)^ C(y)g= W
a=xyf0:5^0:65;0:5^0:3g= 0:5
B C(b) = W
b=xyf B(x)^ C(y)g= W
b=xyf0:4^0:65;0:4^0:3g= 0:4
B C(0) = V
0=xyf B(x)_ C(y)g= V
0=xyf0_0:3;0_0:35;0_0:7;0:5_0:3;0:4_0:3g= 0:3
B C(a) = V
a=xyf B(x)_ C(y)g= V
a=xyf0:5_0:35;0:5_0:7g= 0:5
B C(b) = V
b=xyf B(x)_ C(y)g= V
b=xyf0:4_0:35;0:4_0:7g= 0:4
Thus B C A and B C A, that is B C Abut neither B Anor C A. ThusAis not intuitionistic fuzzy prime bi-ideal ofS.
5.12. Example. Consider the semigroupS=f0; x;1g 0 x 1
0 0 0 0 x 0 x x 1 0 x 1
It is evident thatS is both regular and intra regular.
Bi-ideals inS aref0g;f0; xgandS.
All bi-ideals are strongly prime.
Fact 1. Each intuitionistic fuzzy subset of S is an intuitionistic fuzzy sub- semigroup ofS.
Fact 2. Each intuitionistic fuzzy subsetA= ( A; A)ofSis an intuitionistic fuzzy bi-ideal of S if and only if A(0) A(x) A(1) and A(0) A(x)
A(1).
Proof. SupposeA= ( A; A)is an intuitionistic fuzzy bi-ideal ofS, then by de…nition
A(0) = A(x0x) A(x)^ A(x) = A(x)and
A(0) = A(x0x) A(x)_ A(x) = A(x). Also
A(x) = A(1x1) A(1)^ A(1) = A(1)and
A(x) = A(1x1) A(1)_ A(1) = A(1).
Hence A(0) A(x) A(1)and A(0) A(x) A(1).
Conversely, assume that
A(0) A(x) A(1)and A(0) A(x) A(1).
Then by Fact 1A= ( A; A)is an intuitionistic fuzzy subsemigroup ofS.
Also abc=x if one of a; b; c is xand non of them is 0, and abc = 1 if all of a; b; care1and non of them is 0andabc= 0if one of them is zero. Thus
A(abc) A(a)^ A(c)and A(abc) A(a)_ A(c).
HenceA= ( A; A)is an intuitionistic fuzzy bi-ideal ofS.
Fact 3. Since S is regular and intra regular, so every intuitionistic fuzzy bi-ideal ofS is intuitionistic fuzzy semiprime bi-ideal.
Consider the intuitionistic fuzzy bi-idealsA; B; C ofS,
A(0) = 0:7, A(x) = 0:6, A(0) = 0:5 and A(0) = 0:3, A(x) = 0:4,
A(1) = 0:5
B(0) = 1:0, B(x) = 0:5, B(1) = 0:4 and B(0) = 0, B(x) = 0:5,
B(1) = 0:4
C(0) = 0:7, C(x) = 0:65, C(1) = 0:3 and C(0) = 0:3, C(x) = 0:35,
C(1) = 0:7
B C(0) = W
0=abf B(a)^ C(b)g= W
0=abf1^0:7;1^0:65;1^0:3;0:5^0:7;0:4^0:7g= 0:7
B C(x) = W
x=abf B(a)^ C(b)g= W
x=abf0:5^0:65;0:5^0:3;0:4^0:65g= 0:5
B C(1) = W
1=abf B(a)^ C(b)g= W
1=abf0:4^0:3g= 0:4
B C(0) = V
0=abf B(a)_ C(b)g= V
0=abf0_0:3;0_0:35;0_0:7;0:5_0:3;0:4_0:3g= 0:3
B C(x) = V
x=abf B(a)_ C(b)g= V
x=abf0:5_0:35;0:5_0:7;0:4_0:65g= 0:5
B C(1) = V
1=abf B(a)_ C(b)g= V
1=abf0:4_0:7g= 0:7
Thus B C Aand B C A, that is, B C A, but neitherB Anor C A. ThusAis not intuitionistic fuzzy prime bi-ideal ofS.
5.13. Example. Let S be a group. Then by [21] every intuitionistic fuzzy bi-ideal of S is constant. Since every group is regular and intra regular, so S is regular and intra regular. Since the set of all intuitionistic fuzzy bi-ideals of S is totally ordered, so by Theorem 5.8 every intuitionistic fuzzy bi-ideal ofSis strongly prime.
Acknowledgement 1. The authors wish to thank the anonymous reviewers for their valuable comments which helped to improve this paper.
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