In this paper, we give some theorems which characterizes the intraregular
semigroups in terms of intuitionistic fuzzy left, right, and biideals

Fang, X.

Hong, Yin

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Mathware & Soft Computing 12 (2005) 121-128Characterizing Intraregular Semigroups byIntuitionistic Fuzzy SetsY. Hong1,2 and X. Fang31College of Electronic Information & Engineering,Tongji Univ., 200092, Shanghai, China2College of Information Science & Engineering,Shandong Univ. of Science and Technology,266510, Qingdao, Shandong, China3Maths and Physics Dept.,Anhui Univ. of Science and Technology,232001, Huainan, ChinaAbstractIn this paper, we give some theorems which characterizes the intraregularsemigroups in terms of intuitionistic fuzzy left, right, and biideals.Keywords: Intuitionistic fuzzy ideal; intuitionistic fuzzy biideal; In-traregular semiigroup.1 IntroductionThe concept of fuzzy set was introduced by Zadeh [9]. Fuzzy set theory has beenshown to be a useful tool to describe situations in which the data are imprecise orvague. Fuzzy sets handle such situations by attributing a degree to which a certainobject belongs to a set. The semigroup theory of fuzzy sets was deeply studiedby many authors[1-6]. But in fuzzy sets theory, there is no means to incorporatethe hesitation or uncertainty in the membership degrees. In 1983, Antanassov [10]introduces the concept of intuitionistic fuzzy sets, which constitute a extensionof fuzzy sets theory: intuitionistic fuzzy sets give both a membership degree anda non-membership degree. The only constraint on these two degrees is that thesum must smaller than or equal to 1. In this paper, we give some theorems whichcharacterizes the intraregular semigroups in terms of intuitionistic fuzzy left, right,and biideals.121122 Y. Hong & X. Fang2 PreliminariesLet S be a semigroup, a subsemigroup of S is a nonempty subset A of S suchthat A2 ⊆ A and a left (right) ideal of S is a nonempty subset A of S such thatSA ⊆ A (AS ⊆ A), a two-ideal (or simply ideal) is a subset of S which is both aleft and a right ideal of S.Definition 1[12]: An intuitionistic fuzzy set A in S is an objectA = {〈x, µA (x) , vA (x)〉|x ∈ S}where, for allx ∈ S,µA (x) ∈ [0, 1] and vA (x) ∈ [0, 1] are called the membershipdegree and the non-membership degree, respectively, of x in S, and furthermoresatisfy µA (x) + vA (x) ≤ 1.Definition 2[12]: Let A, B be two intuitionistic fuzzy sets in S, thenA ⊆ B iff (∀x ∈ S) (µA (x) ≤ µB (x) & vA (x) ≤ vB (x)) ,A ∩B = {〈x,min {µA (x) , µB (x)} ,max { vA (x) , vB (x)}〉|x ∈ S} ,A ◦B = {〈x, µA◦B (x) , vA◦B (x)〉|x ∈ S}whereµA◦B (x) ={supx=yz{min {µA (y) , µB (z)}} if x is expressible as x = yz0 otherwise,vA◦B (x) ={infx=yz{max {vA (y) , vB (z)}} if x is expressible as x = yz1 otherwise.Definition 3: If S be a semigroup, an intuitionistic fuzzy set A in S is calledan intuitionistic fuzzy semigroup in S ifµA (xy) ≥ min {µA (x) , µA (y)} and vA (xy) ≤ max {vA (x) , vA (y)} for all x, y ∈S.And is called a intuitionistic fuzzy left (right) ideal of S ifµA (xy) ≥ µA (y) and vA (x, y) ≤ vA (y) (µA (xy) ≥ µA (x) and vA (x, y) ≤ vA (x))for all x, y ∈ S. An intuitionistic fuzzy set A in S is called an intuitionistic fuzzytwo-sided ideal of S if it is both an intuitionistic fuzzy left and an intuitionisticright ideal of S.Characterizing Intraregular Semigroups by Intuitionistic Fuzzy Sets 123Definition 4: An intuitionistic fuzzy semigroupA in S is called an intuitionisticfuzzy biideal of ifµA (xyz) ≥ min {µA (x) , µA (z)} , vA (xyz) ≤ max {vA (x) , vA (z)}for all x, y, z ∈ S.Let A be a subset of a semigroup S, then we denoteA˜ ={〈x, µA˜(x), vA˜(x)〉∣∣∣x ∈ S}whereµA˜(x) ={1 x ∈ A0 otherwise , vA˜(x) ={0 x ∈ A1 otherwise .Obviously A˜ is an intuitionistic fuzzy set in S, semigroup S also can be seen asan intuitionistic fuzzy set S˜ = {〈x, 1, 0〉|x ∈ S}. In the present paper, we will useS represent S and S˜.Definition 5: A semigroup S is called intraregular if, for each element a of S,there exist element x and y of S such that a = xa2y.Lemma 1[13]: For a semigroup S the following conditions are equivalent:(1) S is intraregular;(2) A ∩B ⊂ AB holds for every left ideal A and every right ideal B of S.Lemma 2[15]: A nonempty subset A of a semigroup S is a biideal of S if andonly if A˜ is an intuitionistic fuzzy biideal of S.Lemma 3[15]: For a semigroup S, the following conditions are equivalent:(1) S is regular;(2) A ◦ B = A ∩ B for every intuitionistic fuzzy right ideal A and every intu-itionistic fuzzy left ideal B of S.Lemma 4[15]: For an intuitionistic fuzzy set A of a semigroup S, the followingconditions are equivalent:(1) A is an intuitionistic fuzzy left ideal of S.(2) S ◦A ⊆ A.Lemma 5[15]: For an intuitionistic fuzzy set A of a semigroup S, the followingconditions are equivalent:(1) A is an intuitionistic fuzzy right ideal of S.(2) A ◦ S ⊆ A.Lemma 6[14,15]: Let A be a nonempty subset of a semigroup S, then124 Y. Hong & X. Fang(1) A is a subsemigroup of S if and only if A˜ is an intuitionistic fuzzy semigroupof S.(2) A is a left (right, two-sided) ideal of S if and only if A˜ is an intuitionisticfuzzy left (right, two-sided) of S.3 Characterizing intraregular semigroupsAs is well known, a semigroup S is intraregular if and only if it is a semilatticeof simple semigroups. Now we shall give a characterization of an intraregularsemigroup by intuitionistic fuzzy right ideals and intuitionistic fuzzy left ideals.Theorem 1: For a semigroup S the following conditions are equivalent:(1) S is intraregular;(2) A ∩ B ⊂ B ◦ A holds for every intuitionistic fuzzy right ideal A and everyintuitionistic left ideal B of S.Proof: First assume that (1) holds. Let A be any intuitionistic fuzzy biidealof S, and a any element of S. Since S is intraregular, there exists element x, y inS such that a = xa2y. Then we haveµB◦A (a) = supa=yz{min {µB (y) , µA (z)}}≥ min {µB (ax) , µA (ay)}≥ min {µB (a) , µA (a)}= µA∩B (a)vB◦A (a) = infa=yz{max {vB (y) , vA (z)}}≤ max {vB (ax) , vA (ay)}≤ max {vB (a) , vA (a)}= vA∩B (a)so we have A ∩B ⊆ B ◦A. Thus we obtain (1) implies (2).Conversely, assume (2) holds, let R and L be a right ideal and a left ideal of S,respectively. By Lemma6, R˜ and L˜ be the intuitionistic fuzzy right ideal andintuitionistic fuzzy left ideal of S. Let a be any element of L ∩R, hen we haveµL˜◦R˜ (a) = supa=yz{min{µL˜(y), µR˜(z)}}≥ µL˜◦R˜ (a) ,≥ min{µL˜(a), µR˜(a)}= min{1, 1}= 1Characterizing Intraregular Semigroups by Intuitionistic Fuzzy Sets 125vL˜◦R˜ (a) = infa=yz{max{vL˜(y), vR˜(z)}}≤ vL˜◦R˜ (a)≤ max{vL˜(a), vR˜(a)}= max{0, 0}= 0.This implies that there exist b, c ∈ S, a = bc such thatµL˜(b) = 1, µR˜(c) = 1, vL˜(b) = 0, vR˜(c) = 0.Then we havea = bc ∈ LR,And so we haveL ◦R ⊂ LR.This is follows from Lemma1 that S is intraregular, and (2) implies (1). Thiscompletes the proof.Theorem 2: For a semigroup S the following conditions are equivalent:(1)S is both regular and intraregular;(2)A ◦A = A for every intuitionistic fuzzy biideals A of S;(3)A∩B ⊂ (A ◦B)∩ (B ◦A) for all intuitionistic fuzzy biideals A and B of S;(4) A∩B ⊂ (A ◦B)∩ (B ◦A) for every intuitionistic fuzzy biideal A and everyleft ideal B of S;(5) A∩B ⊂ (A ◦B)∩ (B ◦A) for every intuitionistic fuzzy biideal A and everyintuitionistic fuzzy right ideal B of S;(6) A ∩ B ⊂ (A ◦B) ∩ (B ◦A) for every intuitionistic fuzzy right ideal A andevery intuitionistic fuzzy left ideal B of S.Proof: It is clear that (3) implies (4), (4) implies (6), (3) implies (5), (5) implies(6) , and (3) implies (2). So we will prove that (1) implies (3), (6) implies (1), and(2) implies (1). In the following, we will prove that (1) implies (3), (6) implies (1),and (2) implies (1).First assume that (1) holds. In order to prove that (3) holds, let A and B beany intuitionistic fuzzy biideals of S, and a any element of S. Then, since S isregular, there exists an element x in S such thata = axa(= axaxa).And, since S is intraregular, there exist elements y and z in S such thata = ya2z.126 Y. Hong & X. FangThus we havea = axa = axaxa = ax(ya2z)xa = (axya)(azxa).Since A and B are both intuitionistic fuzzy biideals of S, we haveµA (axya) ≥ min {µA (a) , µA (a)} = µA (a) ,µB (axya) ≥ min {µB (a) , µB (a)} = µB (a) ,AndvA (axya) ≤ max {vA (a) , vA (a)} = vA (a) ,vB (axya) ≤ max {vB (a) , vB (a)} = vB (a) .Thus we haveµA◦B (a) = supa=pq[min {µA (p) , µB (q)}]≥ min {µA (axya) , µB (azxa)}≥ min {µA (a) , µB (a)}= µA∩B (a) ,vA◦B (a) = infa=pq[max {vA (p) , vB (q)}]≤ max {vA (axya) , vB (azxa)}≤ max {vA (a) , vB (a)}= vA∩B (a) .and so we haveA ∩B ⊂ A ◦B.It can be seen in a similar way thatA ∩B ⊂ B ◦A.Thus we obtain thatA ∩B ⊂ (A ◦B) ∩ (B ◦A) ,and that (1) implies (3).Assume that (6) holds. Let A and B be any intuitionistic fuzzy right ideal andany intuitionistic fuzzy left ideal of S, respectively. Then we haveA ∩B ⊂ (A ◦B) ∩ (B ◦A) ⊂ (B ◦A) ,Then it follows from Theorem1 that S is intraregular. On the other hand,A ∩B ⊂ (A ◦B) ∩ (B ◦A) ⊂ (A ◦B) .By Lemma5 and Lemma4 we obtainCharacterizing Intraregular Semigroups by Intuitionistic Fuzzy Sets 127A ◦B ⊆ A ◦ S ⊆ A,A ◦B ⊆ S ◦B ⊆ B,and we haveA ◦B ⊆ A ∩B.Thus we obtain thatA ◦B = A ∩B.Thus it follows from Lemma3 that S is regular. Thus we obtain that (6) implies(1).Finally, assume that (2) holds. In order to prove that (1) holds, let C be anybiideal of S, and a any element of C. Since it follows from lemma that C˜ is anintuitionistic fuzzy biideal of S, we havesupa=pq[min{µC˜(p), µC˜(q)}]= µC˜◦C˜ (a) = µC˜(a) = 1,infa=pq[max{vC˜(p), vC˜(q)}]= vC˜◦C˜ (a) = vC˜(a) = 0.This implies that there exist elements b and c of S with a = bc such thatµC˜(b) = µC˜(c) = 1, vC˜(b) = vC˜(c) = 0.Then we havea = bc ∈ CC,And so we haveC ⊂ CC.Since C is a biideal of S, the converse inclusion always holds. Thus we obtainthatC2 = C.Then it follows from Lemma2 that S is both regular and intraregular. Thereforewe obtain that (2) implies (1). This completes the proof.128 Y. Hong & X. FangReferences:[1] N. Kuroki, Two-sided purity on regular semigroups, Comment. Math. Univ.St. Paul., 24:21-26(1975).[2] N. Kuroki, Fuzzy biideals in semigroup, Comment. Math. Univ. St. Paul.,28:17-21(1979).[3] N. Kuroki, On fuzzy ideals and fuzzy biideals in semigroup, Fuzzy Sets andSystems, 5:203-215(1981).[4] N. Kuroki, Fuzzy semiprime ideals in semigroups, Fuzzy Sets and Systems,8:71-78(1982).[5] N. 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Characterizing intraregular semigroups by intuitionistic fuzzy sets

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