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Fuzzy Generalized Bi-ideals of Γ-semigroups

Fuzzy Generalized Bi-ideals of Γ-semigroups Fuzzy Inf. Eng. (2012) 4: 389-399 DOI 10.1007/s12543-012-0122-0 ORIGINAL ARTICLE S. K. Majumder · M. Mandal Received: 1 March 2012/ Revised: 10 July 2012/ Accepted: 28 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we introduce the concept of fuzzy generalized bi-ideal of a Γ-semigroup, which is an extension of the concept of a fuzzy bi-ideal of a Γ- semigroup and characterize regular Γ-semigroups in terms of fuzzy generalized bi- ideals. Keywords Γ-semigroup · Regular Γ-semigroup · Fuzzy left (right) ideal · Fuzzy ideal· Fuzzy bi-ideal · Fuzzy generalized bi-ideal. 1. Introduction A semigroup is an algebraic structure consisting of a non-empty set S together with an associative binary operation [10]. The formal study of semigroups began in the th early 20 century. Semigroups are important in many areas of mathematics, for ex- ample, coding and language theory, automata theory, combinatorics and mathemat- ical analysis. The concept of fuzzy sets was introduced by L.A. Zadeh [27] in his classic paper in 1965. Azirel Rosenfeld [17] used the idea of fuzzy sets to intro- duce the notions of fuzzy subgroups. Nobuaki Kuroki [12-15] is the pioneer of fuzzy ideal theory of semigroups. The idea of fuzzy subsemigroup was also introduced by Kuroki [12, 14]. In [13], Kuroki characterized several classes of semigroups in terms of fuzzy left, fuzzy right and fuzzy bi-ideals. The notion of fuzzy generalized bi-ideals in semigroups was in introduced by Kuroki [15]. Others, who worked on fuzzy semigroup theory, such as X.Y. Xie [25, 26], Y.B. Jun [11] are mentioned in the bibliography. X.Y. Xie [25] introduced the idea of extensions of fuzzy ideals in semigroups. S. K. Majumder () Tarangapur N.K High School, Tarangapur, Uttar Dinajpur, West Bengal-733129, India email: samitfuzzy@gmail.com M. Mandal () Department of Mathematics, Jadavpur University, Kolkata-700032, India email: manasi ju@yahoo.in − 390 S.K. Majumder· M. Mandal (2012) The notion of aΓ-semigroup was introduced by Sen and Saha [23] as a generaliza- tion of semigroups and ternary semigroup. Γ-semigroup have been analyzed by a lot of mathematicians, for instance by Chattopadhyay [1, 2], Dutta and Adhikari [4, 5], Hila [8, 9], Chinram [3], Saha [21], Sen et al [21-23], Seth [24]. S.K. Sardar and S.K. Majumder [6, 7, 18, 19] have introduced the notion of fuzzification of ideals, prime ideals, semiprime ideals and ideal extensions of Γ-semigroups and studied them via its operator semigroups. In [20], S.K. Sardar, S.K. Majumder and S. Kayal have in- troduced the concepts of fuzzy bi-ideals and fuzzy quasi-ideals in Γ-semigroups and obtained some important characterizations. The purpose of this paper is as stated in the abstract. The structure of the paper is organized as follows. In Section 2, we recall some preliminaries ofΓ-semigroups and fuzzy notions for their use in the sequel. In Section 3, we introduce the notion of generalized bi-ideal and fuzzy generalized bi-ideal of a Γ-semigroup. We use these concepts to characterize regular Γ-semigroup. Theorem 5 characterizes regular Γ-semigroup in terms of generalized bi-ideals and rest of the theorems characterize regular Γ-semigroup in terms of fuzzy generalized bi-ideals. The conclusion is drawn in Section 4. 2. Preliminaries Definition 1 [22] Let S = {x, y, z,···} andΓ= {α,β,γ,···} be two non-empty sets. Then S is called a Γ-semigroup if there exists a mapping S ×Γ× S → S (images to be denoted by aαb) satisfying (i) xγy ∈ S. (ii) (xβy)γz = xβ(yγz),∀x, y, z ∈ S,∀γ ∈ Γ. Example 1 LetΓ= {5, 7}. For any x, y ∈ N and γ ∈ Γ, we define xγy = x.γ.y where “.” is the usual multiplication on N. Then N is a Γ-semigroup. Definition 2 [20] Let S be a Γ-semigroup. By a subsemigroup of S , we mean a non- empty subset A of S such that AΓA ⊆ A. Definition 3 [20] Let S be a Γ-semigroup. By a bi-ideal of S , we mean a subsemi- group A of S such that AΓSΓA ⊆ A. Definition 4 [4] Let S be aΓ-semigroup. By a left (right) ideal of S , we mean a non- empty subset A of S such that SΓA ⊆ A(AΓS ⊆ A). By a two sided ideal or simply an ideal, we mean a non-empty subset of S which is both a left ideal and right ideal of S. Definition 5 [27] A fuzzy subsetμ of a non-empty set X is a function μ : X → [0, 1]. Definition 6 [20] A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy subsemigroup of S if μ(xγy) ≥ min{μ(x),μ(y)}∀x, y ∈ S,∀γ ∈ Γ. Definition 7 [20] A fuzzy subsemigroup μ of a Γ-semigroup S is called a fuzzy bi- ideal of S if μ(xαyβz) ≥ min{μ(x),μ(z)}∀x, y, z ∈ S,∀α,β ∈ Γ. Fuzzy Inf. Eng. (2012) 4: 389-399 391 Definition 8 [18] A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy left ideal of S if μ(xγy) ≥ μ(y) ∀x, y ∈ S, ∀γ ∈ Γ. Definition 9 [18] A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy right ideal of S if μ(xγy) ≥ μ(x) ∀x, y ∈ S,∀γ ∈ Γ. Definition 10 [18] A non-empty fuzzy subset of a Γ-semigroup S is called a fuzzy ideal of S if it is both a fuzzy left ideal and a fuzzy right ideal of S. In the next section, we obtain some important properties of fuzzy generalized bi- ideal of aΓ-semigroup and characterizations of regularΓ-semigroup in terms of fuzzy generalized bi-ideal. 3. Fuzzy Generalized Bi-ideal Definition 11 Let S be a Γ-semigroup. A non-empty subset A of S is called a gener- alized bi-ideal of S if AΓSΓA ⊆ A. Definition 12 A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy gen- eralized bi-ideal of S if μ(xαyβz) ≥ min{μ(x),μ(z)}∀x, y, z ∈ S,∀α,β ∈ Γ. Remark 1 It is clear that fuzzy bi-ideal of S is a subset of fuzzy generalized bi-ideal of S. But in general, the converse inclusion does not hold which will be clear from the following example. Example 2 Let S = {x, y, z, r} andΓ= {γ}, whereγ is defined on S with the following cayley table: γ xy z r x xxxx y xxxx z xx y x r xx yy Then S is a Γ-semigroup. We define a fuzzy subset μ : S → [0, 1] as μ(x) = 0.5, μ(y) = 0,μ(z) = 0.2,μ(r) = 0. Then μ is a fuzzy generalized bi-ideal of S,but μ is not a fuzzy bi-ideal of S. Theorem 1 Let I be a non-empty set of a Γ-semigroup S and χ be the character- istic function of I. Then χ is a fuzzy generalized bi-ideal of S if and only if I is a generalized bi-ideal of S. Proof Let I be a generalized bi-ideal of aΓ-semigroup S andχ be the characteristic function of I. Let x, y, z ∈ S and β, γ ∈ Γ. Then xβyγz ∈ I if x, z ∈ I. It follows that χ (xβyγz) = 1 = min{χ (x),χ (z)}. Let either x  I or z  I. Then I I I Case (i) If xβyγz  I, thenχ (xβyγz) ≥ 0 = min{χ (x),χ (z)}. I I I Case (ii) If xβyγz ∈ I, thenχ (xβyγz) = 1 ≥ 0 = min{χ (x),χ (z)}. I I I Henceχ is a fuzzy generalized bi-ideal of S. Conversely, let χ be a fuzzy generalized bi-ideal of S. Let x, z ∈ I. Then χ (x) = I I χ (z) = 1. Thusχ (xβyγz) ≥ min{χ (x),χ (z)} = 1∀y ∈ S,∀β,γ ∈ Γ. Hence xβyγz ∈ I I I I I ∀y ∈ S,∀β,γ ∈ Γ. Hence I is a generalized bi-ideal of S. 392 S.K. Majumder· M. Mandal (2012) Definition 13 [4] A Γ-semigroup S is called regular if for each element x ∈ S, there exist y ∈ S and α,β ∈ Γ such that x = xαyβx. Proposition 1 Let S be a regularΓ-semigroup. Then every fuzzy generalized bi-ideal of S is a fuzzy bi-ideal S. Proof Let μ be a fuzzy generalized bi-ideal of S. Let a, b ∈ S. Since S is regular, there exist x ∈ S andα,β ∈ Γ such that b = bαxβb. Then for anyγ ∈ Γ, μ(aγb) = μ(aγ(bαxβb)) = μ(aγ(bαx)βb) ≥ min{μ(a),μ(b)}. Soμ is a fuzzy subsemigroup of S and consequentlyμ is a fuzzy bi-ideal of S. Hence the proof. Remark 2 In view of above proposition, we can say that in a regular Γ-semigroup the concept of fuzzy generalized bi-ideal and fuzzy bi-ideal coincide. Proposition 2 Let μ and ν be two fuzzy generalized bi-ideals of a Γ-semigroup S. Thenμ∩ν is a fuzzy generalized bi-ideal of S, providedμ∩ν is non-empty. Proof Let μ and ν be two fuzzy generalized bi-ideals of S and x, y, z ∈ S,α,β ∈ Γ. Then (μ∩ν)(xαyβz) = min{μ(xαyβz),ν(xαyβz)} ≥ min{min{μ(x),μ(z)}, min{ν(x),ν(z)}} = min{min{μ(x),ν(x)}, min{μ(z),ν(z)}} = min{(μ∩ν)(x), (μ∩ν)(z)}. Henceμ∩ν is a fuzzy generalized bi-ideal of S. Definition 14 Let μ and σ be any two fuzzy subsets of a Γ-semigroups S. Then the productμ◦σ is defined as ⎪ sup [min{μ(y),σ(z)} : y, z ∈ S ;γ ∈ Γ], x=yγz (μ◦σ)(x) = 0, otherwise. Lemma 1 Let S be a Γ-semigroup and μ be a non-empty fuzzy subset of S. Then μ is a fuzzy generalized bi-ideal of S if and only if μ ◦ χ ◦ μ ⊆ μ, where χ is the characteristic function of S. Proof Let μ be a fuzzy generalized bi-ideal of S. Suppose there exist x, y, p, q ∈ S and β, γ ∈ Γ such that a = xγy and x = pβq. Since μ is a fuzzy generalized bi-ideal of S, we obtain μ(pβqγy) ≥ min{μ(p),μ(y)}. Then (μ◦χ◦μ)(a) = sup [min{(μ◦χ)(x),μ(y)}] a=xγy Fuzzy Inf. Eng. (2012) 4: 389-399 393 = sup [min{ sup {min{μ(p),χ(q)}},μ(y)}] a=xγy x=pβq = sup [min{ sup {min{μ(p), 1}},μ(y)}] a=xγy x=pβq = sup [min{μ(p),μ(y)}] a=xγy ≤ μ(pβqγy) = μ(xγy) = μ(a). So we have (μ◦χ◦μ) ⊆ μ. Otherwise (μ◦χ◦μ)(a) = 0 ≤ μ(a). Thus (μ◦χ◦μ) ⊆ μ. Conversely, let us assume that μ ◦ χ ◦ μ ⊆ μ. Let x, y, z ∈ S and β, γ ∈ Γ and a = xβyγz. Sinceμ◦χ◦μ ⊆μ, we have μ(xβyγz) = μ(a) ≥ (μ◦χ◦μ)(a) = sup [min{(μ◦χ)(xβy),μ(z)}] a=xβyγz ≥ min{(μ◦χ)(p),μ(z)}(let p = xβy) = min[ sup{min{μ(x),χ(y)}},μ(z)] p=xβy ≥ min[min{μ(x), 1},μ(z)] = min{μ(x),μ(z)}. Henceμ is a fuzzy generalized bi-ideal of S. In view of the above lemma, we have the following theorem. Theorem 2 The product of any two fuzzy generalized bi-ideals of a Γ-semigroup S is a fuzzy generalized bi-ideal of S. Proof Let μ andν be two fuzzy generalized bi-ideals of S. Then (μ◦ν)◦χ◦ (μ◦ν) = μ◦ν◦ (χ◦μ)◦ν ⊆ μ◦ (ν◦χ◦ν) ⊆ μ◦ν. Hence μ◦ν is a fuzzy generalized bi-ideal of S. Similarly, we can show that ν◦μ is also a fuzzy generalized bi-ideal of S. Theorem 3 Let S be a Γ-semigroup. Then following are equivalent: (1) S is regular, (2) For every fuzzy generalized bi-ideal μ of S,μ◦χ◦μ = μ where χ is the charac- teristic function of S. Proof (1) ⇒ (2) Let (1) hold, i.e., S is regular. Let μ be a fuzzy generalized bi-ideal 394 S.K. Majumder· M. Mandal (2012) of S and a ∈ S. Then there exist x ∈ S andα,β ∈ Γ such that a = aαxβa. Hence (μ◦χ◦μ)(a) = sup [min{(μ◦χ)(y),μ(z)}] a=yγz ≥ sup [min{(μ◦χ)(aαx),μ(a)}] a=(aαx)βa ≥ min{(μ◦χ)(aαx),μ(a)} = min[ sup {min{μ(p),χ(q)}},μ(a)](let aαx = pγq) aαx=pγq ≥ min{μ(a),χ(x),μ(a)} = min{μ(a), 1,μ(a)} = μ(a). Soμ ⊆ μ◦χ◦μ. By Lemma 1,μ◦χ◦μ ⊆ μ. Henceμ◦χ◦μ = μ. (2) ⇒ (1) Let us suppose that (2) holds. Let A be a generalized bi-ideal of S. Then by Theorem 1,χ is a fuzzy generalized bi-ideal of S, where χ is the characteristic A A function of A. Hence by hypothesis, χ ◦ χ ◦ χ = χ . Let a ∈ A. Then χ (a) = 1. A A A A Thus (χ ◦χ◦χ )(a) = 1 A A =⇒ sup [min{(χ ◦χ)(b),χ (c)}] = 1 A A a=bγc =⇒ sup [min{ sup min{χ (p),χ(q)},χ (c)}] = 1 A A a=bγc b=pδq =⇒ sup [min{ sup min{χ (p), 1},χ (c)}] = 1 A A a=bγc b=pδq =⇒ sup [min{ supχ (p),χ (c)}] = 1. A A a=bγc b=pδq Thus we get p, c ∈ S such that a = bγc and b = pδq with χ (p) = χ (c) = 1 A A whence p, c ∈ A. So a = bγc = pδqγc ∈ AΓSΓA. Consequently, A ⊆ AΓSΓA. Since A is a generalized bi-ideal of S, so AΓSΓA ⊆ A. Hence A = AΓSΓA and so S is regular. Theorem 4 A Γ-semigroup S is regular if and only if for each fuzzy generalized bi- ideal μ of S and each fuzzy idealν of S,μ∩ν = μ◦ν◦μ. Proof Let S be regular. Let μ be a fuzzy generalized bi-ideal of S and ν be a fuzzy ideal of S. Then by Lemma 1,μ◦ν◦μ ⊆ μ◦χ◦μ ⊆ μ. Again, by Theorem 4.3 [18], μ◦ν◦μ ⊆ χ◦ν◦χ ⊆ χ◦ν ⊆ ν. Soμ◦ν◦μ ⊆ μ∩ν. Now let a ∈ S. Since S is regular, there exist x ∈ S andα,β ∈ Γ such that a = aαxβa = aαxβaαxβa. Then (μ◦ν◦μ)(a) = sup min{μ(y), (ν◦μ)(z)} a=yαz ≥ min{μ(a), (ν◦μ)(xβaαxβa)} Fuzzy Inf. Eng. (2012) 4: 389-399 395 = min[μ(a), sup min{ν(p),μ(q)}] xβaαxβa=pρq ≥ min{μ(a),ν(xβaαx),μ(a)} = min{μ(a),ν(xβaαx)} ≥ min{μ(a),ν(a)}(since ν is a fuzzy ideal of S ) = (μ∩ν)(a). So μ∩ν ⊆ μ◦ν◦μ. Henceμ◦ν◦μ = μ∩ν. Conversely, let us suppose that the necessary condition holds. Let μ be a fuzzy the generalized bi-ideal of S. Then by hypothesis, μ = μ∩χ = μ◦χ◦μ, where χ is characteristic function of S. Hence by Theorem 3, S is regular. Theorem 5 Let S be a Γ-semigroup. Then the following are equivalent: (1) S is regular, (2) A∩ L ⊆ AΓL for each generalized bi-ideal A of S and each left ideal L of S, (3) R∩ A∩ L ⊆ RΓAΓL for each generalized bi-ideal A of S, each left ideal L of S and each right ideal R of S. Proof (1) ⇒ (2) Let S be a regularΓ-semigroup and let a ∈ A∩ L. Then there exist x ∈ S andα,β ∈ Γ such that a = aαxβa. Then a ∈ A and a ∈ L. Since L is a left ideal of S, so xβa ∈ L. This implies that a = aαxβa ∈ AΓL. Hence A∩ L ⊆ AΓL. (1) ⇒ (3) Let S be a regular Γ-semigroup and a ∈ R ∩ A ∩ L. Then there exist x ∈ S and α, β ∈ Γ such that a = aαxβa = aαxβaαxβa. Then a ∈ R and a ∈ A and a ∈ L. Since L is a left ideal and R is a right ideal of S, so xβa ∈ L and aαx ∈ R. This implies that a = aαxβaαxβa ∈ RΓAΓL. Hence R∩ A∩ L ⊆ RΓAΓL. (3) ⇒ (1) Let R∩ A∩ L ⊆ RΓAΓL, for each generalized bi-ideal A of S and for each left ideal L, each right ideal R of S. Let a ∈ S. Let L = {a}∪ SΓa, R = {a}∪ aΓS and A = {a}∪ aΓa∪ aΓSΓa. Then RΓAΓL = ({a}∪ aΓS )Γ({a}∪ aΓa∪ aΓSΓa)Γ({a}∪ SΓa) = (aΓa∪ aΓaΓa∪ aΓaΓSΓa∪ aΓSΓa∪ aΓSΓaΓa∪ aΓSΓaΓSΓa)Γ({a}∪ SΓa) = aΓaΓa∪ aΓaΓSΓa∪ aΓaΓaΓa∪ aΓaΓaΓSΓa∪ aΓaΓSΓaΓa∪ aΓaΓSΓaΓSΓa ∪ aΓSΓaΓa∪ aΓSΓaΓSΓa∪ aΓSΓaΓaΓa∪ aΓSΓaΓaΓSΓa∪ aΓSΓaΓSΓaΓa ∪ aΓSΓaΓSΓaΓSΓa ⊆ aΓSΓa. Since a ∈ R ∩ A ∩ L, then by hypothesis, a ∈ RΓAΓL ⊆ aΓSΓa. So there exist x ∈ S andα,β ∈ Γ such that a = aαxβa. Hence S is regular. (2) ⇒ (1) Let A∩ L ⊆ AΓL for each generalized bi-ideal A of S and for each left ideal L of S. Let a ∈ S. Let L = {a}∪ SΓa and A = {a}∪ aΓa∪ aΓSΓa. Then AΓL = ({a}∪ aΓa∪ aΓSΓa)Γ({a}∪ SΓa) = aΓa∪ aΓSΓa∪ aΓaΓa∪ aΓaΓSΓa∪ aΓSΓaΓa∪ aΓSΓaΓSΓa ⊆ aΓa∪ aΓSΓa. 396 S.K. Majumder· M. Mandal (2012) Since a ∈ A∩ L, then by hypothesis, a ∈ AΓL ⊆ aΓa∪ aΓSΓa. If a ∈ aΓa, then for some α ∈ Γ, a = aαa = aαaαa ∈ aΓSΓa. So there exist x ∈ S and α,β ∈ Γ such that a = aαxβa. Hence S is regular. Theorem 6 Let S be a Γ-semigroup. Then the following are equivalent: (1) S is regular, (2)μ∩ν ⊆ μ◦ν for each fuzzy bi-idealμ of S and for each fuzzy left idealν of S, (3) μ ∩ ν ⊆ μ ◦ ν for each fuzzy generalized bi-ideal μ of S and for each fuzzy left idealν of S, (4)λ∩μ∩ν ⊆ λ◦μ◦ν for each fuzzy bi-idealμ of S, for each fuzzy left idealν of S and for each fuzzy right ideal λ of S, (5)λ∩μ∩ν ⊆ λ◦μ◦ν for each fuzzy generalized bi-idealμ of S and for each fuzzy left ideal ν of S and for each fuzzy right ideal λ of S. Proof (1) ⇒ (2) Let S be regular, μ be a fuzzy bi-ideal of S and ν be a fuzzy left ideal of S. Let a ∈ S. Then there exist x ∈ S and α,β ∈ Γ such that a = aαxβa = aαxβaαxβa. Then (μ◦ν)(a) = sup [min{μ(y),ν(z)}] a=yρz ≥ min{μ(aαxβa),ν(xβa)}(since a = aαxβa = aαxβaαxβa) ≥ min{μ(a),ν(a)}(since μ is a fuzzy bi-ideal of S andν is a fuzzy left ideal of S ) = (μ∩ν)(a). Henceμ◦ν ⊇ μ∩ν. Similarly we can prove that (1) implies (3). (2) ⇒ (1) Let (2) hold, i.e., μ ∩ν ⊆ μ ◦ ν for each fuzzy bi-ideal μ of S and for each fuzzy left ideal ν of S. Since every fuzzy right ideal of S is a fuzzy quasi ideal of S [20] and every fuzzy quasi ideal of S is a fuzzy bi-ideal of S (cf. Proposition 5.2 [20]), soμ∩ν ⊆ μ◦ν for each fuzzy right idealμ of S and for each fuzzy left idealν of S. Alsoμ◦ν ⊆ μ∩ν always holds. Thenμ∩ν = μ◦ν and hence by Theorem 4.7 [18], S is regular. (3) ⇒ (1) Let us suppose that (3) holds. Let A be a generalized bi-ideal of S, L be a left ideal of S and a ∈ A ∩ L. Then a ∈ A and a ∈ L. Since A is a generalized bi-ideal of S, so by Theorem 1,χ is a fuzzy generalized bi-ideal of S, whereχ is the A A characteristic function of A. By Theorem 3.1 [18],χ is a fuzzy left ideal of S, where χ is the characteristic function of L. Hence by hypothesis, χ ∩χ ⊆ χ ◦χ . Then L A L A L (χ ◦χ )(a) ≥ (χ ∩χ )(a) = min{χ (a),χ (a)} = 1. Thus sup [min{χ (y),χ (z)}] = 1. A L A L A L A L a=yγz So there exist b, c ∈ S and δ ∈ Γ with a = bδc such that χ (b) = χ (c) = 1. A L Consequently, b ∈ A and c ∈ L. So a = bδc ∈ AΓL. So A ∩ L ⊆ AΓL. Hence by Theorem 5, S is regular. (1) ⇒ (4) Let S be regular. Let μ be a fuzzy bi-ideal, ν be a fuzzy left ideal and λ be a fuzzy right ideal of S, respectively. Let a ∈ S. Then there exist x ∈ S and Fuzzy Inf. Eng. (2012) 4: 389-399 397 α,β ∈ Γ such that a = aαxβa = aαxβaαxβa = aαxβaαxβaαxβa. Then (λ◦μ◦ν)(a) = sup [min{λ(y), (μ◦ν)(z)}] a=yρz ≥ min{λ(aαx), (μ◦ν)(aαxβaαxβa)} ≥ min{λ(a), (μ◦ν)(aαxβaαxβa)}(since λ is a fuzzy right ideal of S ) = min{λ(a), sup min{μ(aαxβa),ν(xβa)}} (aαxβa)α(xβa))=pρq ≥ min[λ(a), min{μ(aαxβa),ν(xβa)}] ≥ min[λ(a), min{μ(a),ν(a)}](sinceμ is a fuzzy bi-ideal of S andν is a fuzzy left ideal of S ) ≥ min{λ(a),μ(a),ν(a)} = (λ∩μ∩ν)(a). Henceλ∩μ∩ν ⊆ λ◦μ◦ν. Similarly, we can prove that (1) implies (5). (4) ⇒ (1) Let (4) hold. Letλ andν be any fuzzy right ideal and fuzzy left ideal of S, respectively. Since χ is itself a fuzzy bi-ideal of S, where χ is the characteristic S S function of S, by assumption, we have λ∩ν = λ∩χ ∩ν ⊆ λ◦χ ◦ν ⊆ λ◦ν. Also S S λ◦ν ⊆ λ∩ν. Thereforeλ◦ν = λ∩ν. Hence by Theorem 4.7 [18], S is regular. (5) ⇒ (1) Let us suppose that (5) holds. Let A be a generalized bi-ideal of S, L be a left ideal of S, R be a right ideal of S and a ∈ R∩ A∩ L. Then a ∈ R, a ∈ A and a ∈ L. Since A is a generalized bi-ideal of S, so by Theorem 1,χ is a fuzzy generalized bi- ideal of S, where χ is the characteristic function of A. By Theorem 3.1 [18],χ is A L a fuzzy left ideal of S and χ is a fuzzy right ideal of S, where χ and χ are the R L R characteristic functions of L and R respectively. Hence by hypothesis,χ ∩χ ∩χ ⊆ R A L χ ◦χ ◦χ . Then (χ ◦χ ◦χ )(a) ≥ (χ ∩χ ∩χ )(a) = min{χ (a),χ (a),χ (a)} = 1. R A L R A L R A L R A L Thus sup [min{(χ ◦χ )(y),χ (z)}] = 1. R A L a=yγz So there exist b, c ∈ S and δ ∈ Γ with a = bδc such that (χ ◦χ )(b) = χ (c) = 1. R A L Then c ∈ L and sup [min{χ (p),χ (q)}] = 1. Then b = dθe for some d, e ∈ S and R A b=pρq θ ∈ Γ with χ (d) = χ (e) = 1. Consequently, d ∈ R and e ∈ A. So a = bδc = dθeδc ∈ R A RΓAΓL. So R∩ A∩ L ⊆ RΓAΓL. Hence by Theorem 5, S is regular. 4. Conclusion Definition 1 is the definition of one sided Γ-semigroup introduced by M.K. Sen. It may be noted here that in 1981 M.K. Sen [22] introduced the notion of both sided Γ-semigroups, later T.K. Dutta and N.C. Adhikari [4] introduced the notion of both sidedΓ-semigroup and also introduced the notions of operator semigroups of a both sided Γ-semigroup. Throughout this paper, S serves the role of one sided Γ- semigroup. In this paper, the concept of fuzzy generalized bi-ideal of a Γ-semigroup has been introduced and a regular Γ-semigroup has been characterized in terms of fuzzy generalized bi-ideal. Theorems 3-6 illustrate this fact. It is also worthwhile 398 S.K. Majumder· M. Mandal (2012) noting that the corresponding fuzzy generalized bi-ideals of the operator semigroups of a Γ-semigroup may play important roles in furthering the study of the properties of fuzzy generalized bi-ideals of aΓ-semigroup. Acknowledgments The authors are very grateful to Dr. Sujit Kumar Sardar, Associate Professor, Depart- ment of Mathematics, Jadavpur University, Kolkata, India for his inspiration to write this paper. We are also thankful to the learned referees for their valuable comments and suggestions for improving the paper. References 1. Chattopadhyay S (2001) Right inverse Γ-semigroup. Bulletin of Calcutta Mathematical Society 93: 435-442 2. Chattopadhyay S (2005) Right orthodoxΓ-semigroup. South East Asian Bulletin of Mathematics 29: 23-30 3. Chinram R (2006) On quasi-Γ-ideals inΓ-semigroup. Science Asia 32: 351-353 4. Dutta T K, Adhikari N C (1993) On Γ-semigroup with right and left unities. Soochow Journal of Mathematics 19(4): 461-474 5. Dutta T K, Adhikari N C (1994) On prime radical ofΓ-semigroup. Bulletin of Calcutta Mathematical Society 86(5): 437-444 6. Dutta T K, Sardar S K, Majumder S K (2009) Fuzzy ideal extensions of Γ-semigroups. International Mathematical Forum 4(42): 2093-2100 7. Dutta T K, Sardar S K, Majumder S K (2009) Fuzzy ideal extensions ofΓ-semigroups via its operator semigroups. International Journal of Contemporary Mathematical Sciences 4(30): 1455-1463 8. Hila K (2008) On regular, semiprime and quasi-reflexive Γ-semigroup and minimal quasi-ideals. Lobachevskii Journal of Mathematics 29: 141-152 9. Hila K (2007) On some classes of le-Γ-semigroup. Algebras, Groups and Geometries 24: 485-495 10. Howie J M (1995) Fundamentals of semigroup theory. London Mathematical Society Monographs. New Series, 12. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York 11. Jun Y B, Hong S M, Meng J (1995) Fuzzy interior ideals in semigroups. Indian Journal of Pure Applied Mathematics 26(9): 859-863 12. Kuroki N (1981) On fuzzy ideals and fuzzy bi-ideals in semigroups. Fuzzy Sets and Systems 5: 203-215 13. Kuroki N (1991) On fuzzy semigroups. Information Sciences 53: 203-236 14. Kuroki N (1993) Fuzzy semiprime quasi-ideals in semigroups. Information Sciences 75(3): 201-211 15. Kuroki N (1992) Fuzzy generalized bi-ideals in semigroups. Information Sciences 66: 235-243 16. Mordeson et al (2003) Fuzzy semigroups. Springer-Verlag, Heidelberg 17. Rosenfeld A (1971) Fuzzy groups. J. Math. Anal. Appl. 35: 512-517 18. Sardar S K, Majumder S K (2009) On fuzzy ideals inΓ-semigroups. International Journal of Algebra 3(16): 775-784 19. Sardar S K, Majumder S K (2009) A note on characterization of prime ideals of Γ-semigroups in terms of fuzzy subsets. International Journal of Contemporary Mathematical Sciences 4(30): 1465- 20. Sardar S K, Majumder S K, Kayal S (2011) On fuzzy bi-ideals and fuzzy quasi-ideals inΓ-semigroups. “Vasile Alecsandri” University of Bacau ¨ Faculty of Sciences Scientific Studies and Research Series Mathematics and Informatics 21(2): 135-156 21. Saha N K (1987) On Γ-semigroups II. Bulletin of Calcutta Mathematical Society 79: 331-335 22. Sen M K (1981) On Γ-semigroups. Proceedings of the International Conference on Algebra and its Application. Decker Publication, New York: 301-308 23. Sen M K, Saha N K (1986) On Γ-semigroups I. Bulletin of Calcutta Mathematical Society 78: 180- Fuzzy Inf. Eng. (2012) 4: 389-399 399 24. Seth A (1992) Γ-group congruences on regular Γ-semigroups. International Journal of Mathematics and Mathematical Sciences 15(1): 103-106 25. Xie X Y (2001) Fuzzy ideal extensions of semigroups. Soochow Journal of Mathematics 27(2): 125-138 26. Xie X Y (2005) Fuzzy ideal extensions of ordered semigroups. Lobach Journal of Mathematics 19: 29-40 27. Zadeh L A (1965) Fuzzy sets. Information and Control 8: 338-353 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Fuzzy Generalized Bi-ideals of Γ-semigroups

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Fuzzy Inf. Eng. (2012) 4: 389-399 DOI 10.1007/s12543-012-0122-0 ORIGINAL ARTICLE S. K. Majumder · M. Mandal Received: 1 March 2012/ Revised: 10 July 2012/ Accepted: 28 October 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we introduce the concept of fuzzy generalized bi-ideal of a Γ-semigroup, which is an extension of the concept of a fuzzy bi-ideal of a Γ- semigroup and characterize regular Γ-semigroups in terms of fuzzy generalized bi- ideals. Keywords Γ-semigroup · Regular Γ-semigroup · Fuzzy left (right) ideal · Fuzzy ideal· Fuzzy bi-ideal · Fuzzy generalized bi-ideal. 1. Introduction A semigroup is an algebraic structure consisting of a non-empty set S together with an associative binary operation [10]. The formal study of semigroups began in the th early 20 century. Semigroups are important in many areas of mathematics, for ex- ample, coding and language theory, automata theory, combinatorics and mathemat- ical analysis. The concept of fuzzy sets was introduced by L.A. Zadeh [27] in his classic paper in 1965. Azirel Rosenfeld [17] used the idea of fuzzy sets to intro- duce the notions of fuzzy subgroups. Nobuaki Kuroki [12-15] is the pioneer of fuzzy ideal theory of semigroups. The idea of fuzzy subsemigroup was also introduced by Kuroki [12, 14]. In [13], Kuroki characterized several classes of semigroups in terms of fuzzy left, fuzzy right and fuzzy bi-ideals. The notion of fuzzy generalized bi-ideals in semigroups was in introduced by Kuroki [15]. Others, who worked on fuzzy semigroup theory, such as X.Y. Xie [25, 26], Y.B. Jun [11] are mentioned in the bibliography. X.Y. Xie [25] introduced the idea of extensions of fuzzy ideals in semigroups. S. K. Majumder () Tarangapur N.K High School, Tarangapur, Uttar Dinajpur, West Bengal-733129, India email: samitfuzzy@gmail.com M. Mandal () Department of Mathematics, Jadavpur University, Kolkata-700032, India email: manasi ju@yahoo.in − 390 S.K. Majumder· M. Mandal (2012) The notion of aΓ-semigroup was introduced by Sen and Saha [23] as a generaliza- tion of semigroups and ternary semigroup. Γ-semigroup have been analyzed by a lot of mathematicians, for instance by Chattopadhyay [1, 2], Dutta and Adhikari [4, 5], Hila [8, 9], Chinram [3], Saha [21], Sen et al [21-23], Seth [24]. S.K. Sardar and S.K. Majumder [6, 7, 18, 19] have introduced the notion of fuzzification of ideals, prime ideals, semiprime ideals and ideal extensions of Γ-semigroups and studied them via its operator semigroups. In [20], S.K. Sardar, S.K. Majumder and S. Kayal have in- troduced the concepts of fuzzy bi-ideals and fuzzy quasi-ideals in Γ-semigroups and obtained some important characterizations. The purpose of this paper is as stated in the abstract. The structure of the paper is organized as follows. In Section 2, we recall some preliminaries ofΓ-semigroups and fuzzy notions for their use in the sequel. In Section 3, we introduce the notion of generalized bi-ideal and fuzzy generalized bi-ideal of a Γ-semigroup. We use these concepts to characterize regular Γ-semigroup. Theorem 5 characterizes regular Γ-semigroup in terms of generalized bi-ideals and rest of the theorems characterize regular Γ-semigroup in terms of fuzzy generalized bi-ideals. The conclusion is drawn in Section 4. 2. Preliminaries Definition 1 [22] Let S = {x, y, z,···} andΓ= {α,β,γ,···} be two non-empty sets. Then S is called a Γ-semigroup if there exists a mapping S ×Γ× S → S (images to be denoted by aαb) satisfying (i) xγy ∈ S. (ii) (xβy)γz = xβ(yγz),∀x, y, z ∈ S,∀γ ∈ Γ. Example 1 LetΓ= {5, 7}. For any x, y ∈ N and γ ∈ Γ, we define xγy = x.γ.y where “.” is the usual multiplication on N. Then N is a Γ-semigroup. Definition 2 [20] Let S be a Γ-semigroup. By a subsemigroup of S , we mean a non- empty subset A of S such that AΓA ⊆ A. Definition 3 [20] Let S be a Γ-semigroup. By a bi-ideal of S , we mean a subsemi- group A of S such that AΓSΓA ⊆ A. Definition 4 [4] Let S be aΓ-semigroup. By a left (right) ideal of S , we mean a non- empty subset A of S such that SΓA ⊆ A(AΓS ⊆ A). By a two sided ideal or simply an ideal, we mean a non-empty subset of S which is both a left ideal and right ideal of S. Definition 5 [27] A fuzzy subsetμ of a non-empty set X is a function μ : X → [0, 1]. Definition 6 [20] A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy subsemigroup of S if μ(xγy) ≥ min{μ(x),μ(y)}∀x, y ∈ S,∀γ ∈ Γ. Definition 7 [20] A fuzzy subsemigroup μ of a Γ-semigroup S is called a fuzzy bi- ideal of S if μ(xαyβz) ≥ min{μ(x),μ(z)}∀x, y, z ∈ S,∀α,β ∈ Γ. Fuzzy Inf. Eng. (2012) 4: 389-399 391 Definition 8 [18] A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy left ideal of S if μ(xγy) ≥ μ(y) ∀x, y ∈ S, ∀γ ∈ Γ. Definition 9 [18] A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy right ideal of S if μ(xγy) ≥ μ(x) ∀x, y ∈ S,∀γ ∈ Γ. Definition 10 [18] A non-empty fuzzy subset of a Γ-semigroup S is called a fuzzy ideal of S if it is both a fuzzy left ideal and a fuzzy right ideal of S. In the next section, we obtain some important properties of fuzzy generalized bi- ideal of aΓ-semigroup and characterizations of regularΓ-semigroup in terms of fuzzy generalized bi-ideal. 3. Fuzzy Generalized Bi-ideal Definition 11 Let S be a Γ-semigroup. A non-empty subset A of S is called a gener- alized bi-ideal of S if AΓSΓA ⊆ A. Definition 12 A non-empty fuzzy subset μ of a Γ-semigroup S is called a fuzzy gen- eralized bi-ideal of S if μ(xαyβz) ≥ min{μ(x),μ(z)}∀x, y, z ∈ S,∀α,β ∈ Γ. Remark 1 It is clear that fuzzy bi-ideal of S is a subset of fuzzy generalized bi-ideal of S. But in general, the converse inclusion does not hold which will be clear from the following example. Example 2 Let S = {x, y, z, r} andΓ= {γ}, whereγ is defined on S with the following cayley table: γ xy z r x xxxx y xxxx z xx y x r xx yy Then S is a Γ-semigroup. We define a fuzzy subset μ : S → [0, 1] as μ(x) = 0.5, μ(y) = 0,μ(z) = 0.2,μ(r) = 0. Then μ is a fuzzy generalized bi-ideal of S,but μ is not a fuzzy bi-ideal of S. Theorem 1 Let I be a non-empty set of a Γ-semigroup S and χ be the character- istic function of I. Then χ is a fuzzy generalized bi-ideal of S if and only if I is a generalized bi-ideal of S. Proof Let I be a generalized bi-ideal of aΓ-semigroup S andχ be the characteristic function of I. Let x, y, z ∈ S and β, γ ∈ Γ. Then xβyγz ∈ I if x, z ∈ I. It follows that χ (xβyγz) = 1 = min{χ (x),χ (z)}. Let either x  I or z  I. Then I I I Case (i) If xβyγz  I, thenχ (xβyγz) ≥ 0 = min{χ (x),χ (z)}. I I I Case (ii) If xβyγz ∈ I, thenχ (xβyγz) = 1 ≥ 0 = min{χ (x),χ (z)}. I I I Henceχ is a fuzzy generalized bi-ideal of S. Conversely, let χ be a fuzzy generalized bi-ideal of S. Let x, z ∈ I. Then χ (x) = I I χ (z) = 1. Thusχ (xβyγz) ≥ min{χ (x),χ (z)} = 1∀y ∈ S,∀β,γ ∈ Γ. Hence xβyγz ∈ I I I I I ∀y ∈ S,∀β,γ ∈ Γ. Hence I is a generalized bi-ideal of S. 392 S.K. Majumder· M. Mandal (2012) Definition 13 [4] A Γ-semigroup S is called regular if for each element x ∈ S, there exist y ∈ S and α,β ∈ Γ such that x = xαyβx. Proposition 1 Let S be a regularΓ-semigroup. Then every fuzzy generalized bi-ideal of S is a fuzzy bi-ideal S. Proof Let μ be a fuzzy generalized bi-ideal of S. Let a, b ∈ S. Since S is regular, there exist x ∈ S andα,β ∈ Γ such that b = bαxβb. Then for anyγ ∈ Γ, μ(aγb) = μ(aγ(bαxβb)) = μ(aγ(bαx)βb) ≥ min{μ(a),μ(b)}. Soμ is a fuzzy subsemigroup of S and consequentlyμ is a fuzzy bi-ideal of S. Hence the proof. Remark 2 In view of above proposition, we can say that in a regular Γ-semigroup the concept of fuzzy generalized bi-ideal and fuzzy bi-ideal coincide. Proposition 2 Let μ and ν be two fuzzy generalized bi-ideals of a Γ-semigroup S. Thenμ∩ν is a fuzzy generalized bi-ideal of S, providedμ∩ν is non-empty. Proof Let μ and ν be two fuzzy generalized bi-ideals of S and x, y, z ∈ S,α,β ∈ Γ. Then (μ∩ν)(xαyβz) = min{μ(xαyβz),ν(xαyβz)} ≥ min{min{μ(x),μ(z)}, min{ν(x),ν(z)}} = min{min{μ(x),ν(x)}, min{μ(z),ν(z)}} = min{(μ∩ν)(x), (μ∩ν)(z)}. Henceμ∩ν is a fuzzy generalized bi-ideal of S. Definition 14 Let μ and σ be any two fuzzy subsets of a Γ-semigroups S. Then the productμ◦σ is defined as ⎪ sup [min{μ(y),σ(z)} : y, z ∈ S ;γ ∈ Γ], x=yγz (μ◦σ)(x) = 0, otherwise. Lemma 1 Let S be a Γ-semigroup and μ be a non-empty fuzzy subset of S. Then μ is a fuzzy generalized bi-ideal of S if and only if μ ◦ χ ◦ μ ⊆ μ, where χ is the characteristic function of S. Proof Let μ be a fuzzy generalized bi-ideal of S. Suppose there exist x, y, p, q ∈ S and β, γ ∈ Γ such that a = xγy and x = pβq. Since μ is a fuzzy generalized bi-ideal of S, we obtain μ(pβqγy) ≥ min{μ(p),μ(y)}. Then (μ◦χ◦μ)(a) = sup [min{(μ◦χ)(x),μ(y)}] a=xγy Fuzzy Inf. Eng. (2012) 4: 389-399 393 = sup [min{ sup {min{μ(p),χ(q)}},μ(y)}] a=xγy x=pβq = sup [min{ sup {min{μ(p), 1}},μ(y)}] a=xγy x=pβq = sup [min{μ(p),μ(y)}] a=xγy ≤ μ(pβqγy) = μ(xγy) = μ(a). So we have (μ◦χ◦μ) ⊆ μ. Otherwise (μ◦χ◦μ)(a) = 0 ≤ μ(a). Thus (μ◦χ◦μ) ⊆ μ. Conversely, let us assume that μ ◦ χ ◦ μ ⊆ μ. Let x, y, z ∈ S and β, γ ∈ Γ and a = xβyγz. Sinceμ◦χ◦μ ⊆μ, we have μ(xβyγz) = μ(a) ≥ (μ◦χ◦μ)(a) = sup [min{(μ◦χ)(xβy),μ(z)}] a=xβyγz ≥ min{(μ◦χ)(p),μ(z)}(let p = xβy) = min[ sup{min{μ(x),χ(y)}},μ(z)] p=xβy ≥ min[min{μ(x), 1},μ(z)] = min{μ(x),μ(z)}. Henceμ is a fuzzy generalized bi-ideal of S. In view of the above lemma, we have the following theorem. Theorem 2 The product of any two fuzzy generalized bi-ideals of a Γ-semigroup S is a fuzzy generalized bi-ideal of S. Proof Let μ andν be two fuzzy generalized bi-ideals of S. Then (μ◦ν)◦χ◦ (μ◦ν) = μ◦ν◦ (χ◦μ)◦ν ⊆ μ◦ (ν◦χ◦ν) ⊆ μ◦ν. Hence μ◦ν is a fuzzy generalized bi-ideal of S. Similarly, we can show that ν◦μ is also a fuzzy generalized bi-ideal of S. Theorem 3 Let S be a Γ-semigroup. Then following are equivalent: (1) S is regular, (2) For every fuzzy generalized bi-ideal μ of S,μ◦χ◦μ = μ where χ is the charac- teristic function of S. Proof (1) ⇒ (2) Let (1) hold, i.e., S is regular. Let μ be a fuzzy generalized bi-ideal 394 S.K. Majumder· M. Mandal (2012) of S and a ∈ S. Then there exist x ∈ S andα,β ∈ Γ such that a = aαxβa. Hence (μ◦χ◦μ)(a) = sup [min{(μ◦χ)(y),μ(z)}] a=yγz ≥ sup [min{(μ◦χ)(aαx),μ(a)}] a=(aαx)βa ≥ min{(μ◦χ)(aαx),μ(a)} = min[ sup {min{μ(p),χ(q)}},μ(a)](let aαx = pγq) aαx=pγq ≥ min{μ(a),χ(x),μ(a)} = min{μ(a), 1,μ(a)} = μ(a). Soμ ⊆ μ◦χ◦μ. By Lemma 1,μ◦χ◦μ ⊆ μ. Henceμ◦χ◦μ = μ. (2) ⇒ (1) Let us suppose that (2) holds. Let A be a generalized bi-ideal of S. Then by Theorem 1,χ is a fuzzy generalized bi-ideal of S, where χ is the characteristic A A function of A. Hence by hypothesis, χ ◦ χ ◦ χ = χ . Let a ∈ A. Then χ (a) = 1. A A A A Thus (χ ◦χ◦χ )(a) = 1 A A =⇒ sup [min{(χ ◦χ)(b),χ (c)}] = 1 A A a=bγc =⇒ sup [min{ sup min{χ (p),χ(q)},χ (c)}] = 1 A A a=bγc b=pδq =⇒ sup [min{ sup min{χ (p), 1},χ (c)}] = 1 A A a=bγc b=pδq =⇒ sup [min{ supχ (p),χ (c)}] = 1. A A a=bγc b=pδq Thus we get p, c ∈ S such that a = bγc and b = pδq with χ (p) = χ (c) = 1 A A whence p, c ∈ A. So a = bγc = pδqγc ∈ AΓSΓA. Consequently, A ⊆ AΓSΓA. Since A is a generalized bi-ideal of S, so AΓSΓA ⊆ A. Hence A = AΓSΓA and so S is regular. Theorem 4 A Γ-semigroup S is regular if and only if for each fuzzy generalized bi- ideal μ of S and each fuzzy idealν of S,μ∩ν = μ◦ν◦μ. Proof Let S be regular. Let μ be a fuzzy generalized bi-ideal of S and ν be a fuzzy ideal of S. Then by Lemma 1,μ◦ν◦μ ⊆ μ◦χ◦μ ⊆ μ. Again, by Theorem 4.3 [18], μ◦ν◦μ ⊆ χ◦ν◦χ ⊆ χ◦ν ⊆ ν. Soμ◦ν◦μ ⊆ μ∩ν. Now let a ∈ S. Since S is regular, there exist x ∈ S andα,β ∈ Γ such that a = aαxβa = aαxβaαxβa. Then (μ◦ν◦μ)(a) = sup min{μ(y), (ν◦μ)(z)} a=yαz ≥ min{μ(a), (ν◦μ)(xβaαxβa)} Fuzzy Inf. Eng. (2012) 4: 389-399 395 = min[μ(a), sup min{ν(p),μ(q)}] xβaαxβa=pρq ≥ min{μ(a),ν(xβaαx),μ(a)} = min{μ(a),ν(xβaαx)} ≥ min{μ(a),ν(a)}(since ν is a fuzzy ideal of S ) = (μ∩ν)(a). So μ∩ν ⊆ μ◦ν◦μ. Henceμ◦ν◦μ = μ∩ν. Conversely, let us suppose that the necessary condition holds. Let μ be a fuzzy the generalized bi-ideal of S. Then by hypothesis, μ = μ∩χ = μ◦χ◦μ, where χ is characteristic function of S. Hence by Theorem 3, S is regular. Theorem 5 Let S be a Γ-semigroup. Then the following are equivalent: (1) S is regular, (2) A∩ L ⊆ AΓL for each generalized bi-ideal A of S and each left ideal L of S, (3) R∩ A∩ L ⊆ RΓAΓL for each generalized bi-ideal A of S, each left ideal L of S and each right ideal R of S. Proof (1) ⇒ (2) Let S be a regularΓ-semigroup and let a ∈ A∩ L. Then there exist x ∈ S andα,β ∈ Γ such that a = aαxβa. Then a ∈ A and a ∈ L. Since L is a left ideal of S, so xβa ∈ L. This implies that a = aαxβa ∈ AΓL. Hence A∩ L ⊆ AΓL. (1) ⇒ (3) Let S be a regular Γ-semigroup and a ∈ R ∩ A ∩ L. Then there exist x ∈ S and α, β ∈ Γ such that a = aαxβa = aαxβaαxβa. Then a ∈ R and a ∈ A and a ∈ L. Since L is a left ideal and R is a right ideal of S, so xβa ∈ L and aαx ∈ R. This implies that a = aαxβaαxβa ∈ RΓAΓL. Hence R∩ A∩ L ⊆ RΓAΓL. (3) ⇒ (1) Let R∩ A∩ L ⊆ RΓAΓL, for each generalized bi-ideal A of S and for each left ideal L, each right ideal R of S. Let a ∈ S. Let L = {a}∪ SΓa, R = {a}∪ aΓS and A = {a}∪ aΓa∪ aΓSΓa. Then RΓAΓL = ({a}∪ aΓS )Γ({a}∪ aΓa∪ aΓSΓa)Γ({a}∪ SΓa) = (aΓa∪ aΓaΓa∪ aΓaΓSΓa∪ aΓSΓa∪ aΓSΓaΓa∪ aΓSΓaΓSΓa)Γ({a}∪ SΓa) = aΓaΓa∪ aΓaΓSΓa∪ aΓaΓaΓa∪ aΓaΓaΓSΓa∪ aΓaΓSΓaΓa∪ aΓaΓSΓaΓSΓa ∪ aΓSΓaΓa∪ aΓSΓaΓSΓa∪ aΓSΓaΓaΓa∪ aΓSΓaΓaΓSΓa∪ aΓSΓaΓSΓaΓa ∪ aΓSΓaΓSΓaΓSΓa ⊆ aΓSΓa. Since a ∈ R ∩ A ∩ L, then by hypothesis, a ∈ RΓAΓL ⊆ aΓSΓa. So there exist x ∈ S andα,β ∈ Γ such that a = aαxβa. Hence S is regular. (2) ⇒ (1) Let A∩ L ⊆ AΓL for each generalized bi-ideal A of S and for each left ideal L of S. Let a ∈ S. Let L = {a}∪ SΓa and A = {a}∪ aΓa∪ aΓSΓa. Then AΓL = ({a}∪ aΓa∪ aΓSΓa)Γ({a}∪ SΓa) = aΓa∪ aΓSΓa∪ aΓaΓa∪ aΓaΓSΓa∪ aΓSΓaΓa∪ aΓSΓaΓSΓa ⊆ aΓa∪ aΓSΓa. 396 S.K. Majumder· M. Mandal (2012) Since a ∈ A∩ L, then by hypothesis, a ∈ AΓL ⊆ aΓa∪ aΓSΓa. If a ∈ aΓa, then for some α ∈ Γ, a = aαa = aαaαa ∈ aΓSΓa. So there exist x ∈ S and α,β ∈ Γ such that a = aαxβa. Hence S is regular. Theorem 6 Let S be a Γ-semigroup. Then the following are equivalent: (1) S is regular, (2)μ∩ν ⊆ μ◦ν for each fuzzy bi-idealμ of S and for each fuzzy left idealν of S, (3) μ ∩ ν ⊆ μ ◦ ν for each fuzzy generalized bi-ideal μ of S and for each fuzzy left idealν of S, (4)λ∩μ∩ν ⊆ λ◦μ◦ν for each fuzzy bi-idealμ of S, for each fuzzy left idealν of S and for each fuzzy right ideal λ of S, (5)λ∩μ∩ν ⊆ λ◦μ◦ν for each fuzzy generalized bi-idealμ of S and for each fuzzy left ideal ν of S and for each fuzzy right ideal λ of S. Proof (1) ⇒ (2) Let S be regular, μ be a fuzzy bi-ideal of S and ν be a fuzzy left ideal of S. Let a ∈ S. Then there exist x ∈ S and α,β ∈ Γ such that a = aαxβa = aαxβaαxβa. Then (μ◦ν)(a) = sup [min{μ(y),ν(z)}] a=yρz ≥ min{μ(aαxβa),ν(xβa)}(since a = aαxβa = aαxβaαxβa) ≥ min{μ(a),ν(a)}(since μ is a fuzzy bi-ideal of S andν is a fuzzy left ideal of S ) = (μ∩ν)(a). Henceμ◦ν ⊇ μ∩ν. Similarly we can prove that (1) implies (3). (2) ⇒ (1) Let (2) hold, i.e., μ ∩ν ⊆ μ ◦ ν for each fuzzy bi-ideal μ of S and for each fuzzy left ideal ν of S. Since every fuzzy right ideal of S is a fuzzy quasi ideal of S [20] and every fuzzy quasi ideal of S is a fuzzy bi-ideal of S (cf. Proposition 5.2 [20]), soμ∩ν ⊆ μ◦ν for each fuzzy right idealμ of S and for each fuzzy left idealν of S. Alsoμ◦ν ⊆ μ∩ν always holds. Thenμ∩ν = μ◦ν and hence by Theorem 4.7 [18], S is regular. (3) ⇒ (1) Let us suppose that (3) holds. Let A be a generalized bi-ideal of S, L be a left ideal of S and a ∈ A ∩ L. Then a ∈ A and a ∈ L. Since A is a generalized bi-ideal of S, so by Theorem 1,χ is a fuzzy generalized bi-ideal of S, whereχ is the A A characteristic function of A. By Theorem 3.1 [18],χ is a fuzzy left ideal of S, where χ is the characteristic function of L. Hence by hypothesis, χ ∩χ ⊆ χ ◦χ . Then L A L A L (χ ◦χ )(a) ≥ (χ ∩χ )(a) = min{χ (a),χ (a)} = 1. Thus sup [min{χ (y),χ (z)}] = 1. A L A L A L A L a=yγz So there exist b, c ∈ S and δ ∈ Γ with a = bδc such that χ (b) = χ (c) = 1. A L Consequently, b ∈ A and c ∈ L. So a = bδc ∈ AΓL. So A ∩ L ⊆ AΓL. Hence by Theorem 5, S is regular. (1) ⇒ (4) Let S be regular. Let μ be a fuzzy bi-ideal, ν be a fuzzy left ideal and λ be a fuzzy right ideal of S, respectively. Let a ∈ S. Then there exist x ∈ S and Fuzzy Inf. Eng. (2012) 4: 389-399 397 α,β ∈ Γ such that a = aαxβa = aαxβaαxβa = aαxβaαxβaαxβa. Then (λ◦μ◦ν)(a) = sup [min{λ(y), (μ◦ν)(z)}] a=yρz ≥ min{λ(aαx), (μ◦ν)(aαxβaαxβa)} ≥ min{λ(a), (μ◦ν)(aαxβaαxβa)}(since λ is a fuzzy right ideal of S ) = min{λ(a), sup min{μ(aαxβa),ν(xβa)}} (aαxβa)α(xβa))=pρq ≥ min[λ(a), min{μ(aαxβa),ν(xβa)}] ≥ min[λ(a), min{μ(a),ν(a)}](sinceμ is a fuzzy bi-ideal of S andν is a fuzzy left ideal of S ) ≥ min{λ(a),μ(a),ν(a)} = (λ∩μ∩ν)(a). Henceλ∩μ∩ν ⊆ λ◦μ◦ν. Similarly, we can prove that (1) implies (5). (4) ⇒ (1) Let (4) hold. Letλ andν be any fuzzy right ideal and fuzzy left ideal of S, respectively. Since χ is itself a fuzzy bi-ideal of S, where χ is the characteristic S S function of S, by assumption, we have λ∩ν = λ∩χ ∩ν ⊆ λ◦χ ◦ν ⊆ λ◦ν. Also S S λ◦ν ⊆ λ∩ν. Thereforeλ◦ν = λ∩ν. Hence by Theorem 4.7 [18], S is regular. (5) ⇒ (1) Let us suppose that (5) holds. Let A be a generalized bi-ideal of S, L be a left ideal of S, R be a right ideal of S and a ∈ R∩ A∩ L. Then a ∈ R, a ∈ A and a ∈ L. Since A is a generalized bi-ideal of S, so by Theorem 1,χ is a fuzzy generalized bi- ideal of S, where χ is the characteristic function of A. By Theorem 3.1 [18],χ is A L a fuzzy left ideal of S and χ is a fuzzy right ideal of S, where χ and χ are the R L R characteristic functions of L and R respectively. Hence by hypothesis,χ ∩χ ∩χ ⊆ R A L χ ◦χ ◦χ . Then (χ ◦χ ◦χ )(a) ≥ (χ ∩χ ∩χ )(a) = min{χ (a),χ (a),χ (a)} = 1. R A L R A L R A L R A L Thus sup [min{(χ ◦χ )(y),χ (z)}] = 1. R A L a=yγz So there exist b, c ∈ S and δ ∈ Γ with a = bδc such that (χ ◦χ )(b) = χ (c) = 1. R A L Then c ∈ L and sup [min{χ (p),χ (q)}] = 1. Then b = dθe for some d, e ∈ S and R A b=pρq θ ∈ Γ with χ (d) = χ (e) = 1. Consequently, d ∈ R and e ∈ A. So a = bδc = dθeδc ∈ R A RΓAΓL. So R∩ A∩ L ⊆ RΓAΓL. Hence by Theorem 5, S is regular. 4. Conclusion Definition 1 is the definition of one sided Γ-semigroup introduced by M.K. Sen. It may be noted here that in 1981 M.K. Sen [22] introduced the notion of both sided Γ-semigroups, later T.K. Dutta and N.C. Adhikari [4] introduced the notion of both sidedΓ-semigroup and also introduced the notions of operator semigroups of a both sided Γ-semigroup. Throughout this paper, S serves the role of one sided Γ- semigroup. In this paper, the concept of fuzzy generalized bi-ideal of a Γ-semigroup has been introduced and a regular Γ-semigroup has been characterized in terms of fuzzy generalized bi-ideal. Theorems 3-6 illustrate this fact. It is also worthwhile 398 S.K. Majumder· M. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Dec 1, 2012

Keywords: Γ-semigroup; Regular Γ-semigroup; Fuzzy left (right) ideal; Fuzzy ideal; Fuzzy bi-ideal; Fuzzy generalized bi-ideal

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