Fuzzy Generalized Bi-ideals of Γ-Semigroups Samit Kumar Majumder1 and Manasi Mandal2 1
Tarangapur N.K High School, Tarangapur, Uttar Dinajpur, West Bengal-733129, INDIA 2 Department of Mathematics, Jadavpur University, Kolkata-700032, INDIA 1
[email protected] 2 manasi−
[email protected]
Abstract In this paper we introduce the concept of fuzzy generalized bi-ideal of a Γsemigroup, 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. AMS Mathematics Subject Classification(2000): 08A72, 20M12, 3F55 Key Words and Phrases: 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 early 20th century. Semigroups are important in many areas of mathematics, for example, coding and language theory, automata theory, combinatorics and mathematical analysis. The concept of fuzzy sets was introduced by Lofti Zadeh[27] in his classic paper in 1965. Azirel Rosenfeld[17] used the idea of fuzzy set to introduce the notions of fuzzy subgroups. Nobuaki Kuroki[12, 13, 14, 16] 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. 1
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. The notion of a Γ-semigroup was introduced by Sen and Saha[23] as a generalization of semigroups and ternary semigroup. Γ-semigroup have been analyzed by lot of mathematicians, for instance by Chattopadhyay[1, 2], Dutta and Adhikari[4, 5], Hila[8, 9], Chinram[3], Saha[21], Sen et al.[22, 23, 21], 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 introduced 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.
2
Preliminaries
In this section we discuss some elementary definitions that we use in the sequel. Definition 2.1. [22] Let S = {x, y, z, . . .} and Γ = {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 (1) xγy ∈ S, (2) (xβy)γz = xβ(yγz), ∀x, y, z ∈ S, ∀γ ∈ Γ. Example 1. Let Γ = {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 2.2. [20] Let S be a Γ-semigroup. By a subsemigroup(in short SS) of S we mean a non-empty subset A of S such that AΓA ⊆ A. Definition 2.3. [20] Let S be a Γ-semigroup. By a bi-ideal(in short BI) of S we mean a SS A of S such that AΓSΓA ⊆ A. Definition 2.4. [4] Let S be a Γ-semigroup. By a left(right) ideal(in short LI(RI)) 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(in short I), we mean a non-empty subset of S which is both a left ideal(LI) and right ideal(RI) of S. Definition 2.5. [27] A fuzzy subset µ of a non-empty set X is a function µ : X → [0, 1]. Definition 2.6. [20] A non-empty fuzzy subset µ of a Γ-semigroup S is called a fuzzy subsemigroup(in short FSS) of S if µ(xγy) ≥ min{µ(x), µ(y)} ∀x, y ∈ S, ∀γ ∈ Γ. Definition 2.7. [20] A FSS µ of a Γ-semigroup S is called a fuzzy bi-ideal( in short FBI) of S if µ(xαyβz) ≥ min{µ(x), µ(z)} ∀x, y, z ∈ S, ∀α, β ∈ Γ.
2
Definition 2.8. [18] A non-empty fuzzy subset µ of a Γ-semigroup S is called a fuzzy left ideal( in short FLI) of S if µ(xγy) ≥ µ(y) ∀x, y ∈ S, ∀γ ∈ Γ. Definition 2.9. [18] A non-empty fuzzy subset µ of a Γ-semigroup S is called a fuzzy right ideal( in short FRI) of S if µ(xγy) ≥ µ(x) ∀x, y ∈ S, ∀γ ∈ Γ. Definition 2.10. [18] A non-empty fuzzy subset of a Γ-semigroup S is called a fuzzy ideal( in short FI) of S if it is both a fuzzy left ideal(FLI) and a fuzzy right ideal(FRI) of S.
3
Fuzzy Generalized Bi-ideal
Definition 3.1. Let S be a Γ-semigroup. a non-empty subset A of S is called a generalized bi-ideal(in short GBI) of S if AΓSΓA ⊆ A. Definition 3.2. A non-empty fuzzy subset µ of a Γ-semigroup S is called a fuzzy generalized bi-ideal(in short FGBI) of S if µ(xαyβz) ≥ min{µ(x), µ(z)} ∀x, y, z ∈ S, ∀α, β ∈ Γ. Remark 1. It is clear that FBI(S) ⊆ FGBI(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} and Γ following cayley table: γ x x x y x z x r x
= {γ}, where γ is defined on S with the y x x x x
z x x y y
r x x x y
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 FGBI of S but µ is not a FBI of S. We obtain the following theorem by routine verification. Theorem 3.3. Let I be a non-empty set of a Γ-semigroup S and χI be the characteristic function of I. Then χI is a FGBI of S if and only if I is a GBI of S. Definition 3.4. [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 3.5. Let S be a regular Γ-semigroup. Then FGBI(S) ⊆ FBI(S). Proof. Let µ be a FGBI of S. Let a, b ∈ S. Since S is regular, there exist x ∈ S and α, β ∈ Γ such that b = bαxβb. Then for any γ ∈ Γ, µ(aγb) = µ(aγ(bαxβb)) = µ(aγ(bαx)βb) ≥ min{µ(a), µ(b)}. So µ is a FSS of S and consequently µ is a FBI of S. Hence FGBI(S) ⊆ FBI(S). 3
Remark 2. In view of above proposition we can say that in a regular Γ-semigroup the concept of FGBI and FBI coincide. We obtain the following proposition by routine verification. Proposition 3.6. Let µ and ν be two FGBIs of a Γ-semigroup S. Then µ ∩ ν is a FGBI of S, provided µ ∩ ν is non-empty. Definition 3.7. Let µ and σ be any two fuzzy subsets of a Γ-semigroups S. Then the product µ ◦ σ is ( defined as sup [min{µ(y), σ(z)}] x=yγz (µ ◦ σ)(x) = . 0, Otherwise We obtain the following lemma by routine verification. Lemma 3.8. Let S be a Γ-semigroup and µ be a non-empty fuzzy subset of S. Then µ is a FGBI of S if and only if µ ◦ χ ◦ µ ⊆ µ, where χ is the characteristic function of S. In view of the above lemma we have the following theorem. Theorem 3.9. The product of any two FGBIs of a Γ-semigroup S is a FGBI of S. Proof. Let µ and ν be two FGBIs of S. Then (µ ◦ ν) ◦ χ ◦ (µ ◦ ν) = µ ◦ ν ◦ (χ ◦ µ) ◦ ν ⊆ µ ◦ (ν ◦ χ ◦ ν) ⊆ µ ◦ ν. Hence µ ◦ ν is a FGBI of S. Similarly we can show that ν ◦ µ is also a FGBI of S. Theorem 3.10. Let S be a Γ-semigroup. Then following are equivalent: (1) S is regular, (2) for every FGBI µ of S, µ ◦ χ ◦ µ = µ where χ is the characteristic function of S. Proof. (1) ⇒ (2) : Let (1) holds, i.e., S is regular. Let µ be a FGBI of S and a ∈ S. Then there exist x ∈ S and α, β ∈ Γ such that a = aαxβa. Then (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). So µ ⊆ µ ◦ χ ◦ µ. By Lemma 3.8, µ ◦ χ ◦ µ ⊆ µ. Hence µ ◦ χ ◦ µ = µ.
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(2) ⇒ (1) : Let us suppose that (2) holds. Let A be a GBI of S. Then by Theorem 3.3, χA is a FGBI of S, where χA is the characteristic function of A. Then by hypothesis χA ◦ χ ◦ χA = χA . Let a ∈ A. Then χA (a) = 1. Thus (χA ◦ χ ◦ χA )(a) = 1 =⇒ sup [min{(χA ◦ χ)(b), χA (c)}] = 1 a=bγc
=⇒ sup [min{ sup min{χA (p), χ(q)}, χA (c)}] = 1 a=bγc
b=pδq
=⇒ sup [min{ sup min{χA (p), 1}, χA (c)}] = 1 a=bγc
b=pδq
=⇒ sup [min{ sup χA (p), χA (c)}] = 1. a=bγc
b=pδq
Thus χA (p) = χA (c) = 1 and hence p, c ∈ A. So a = bγc = pδqγc ∈ AΓSΓA. Consequently, A ⊆ AΓSΓA. Since A is a GBI of S so AΓSΓA ⊆ A. Hence A = AΓSΓA and so S is regular. Theorem 3.11. A Γ-semigroup S is regular if and only if for each FGBI, µ of S and each FI, ν of S, µ ∩ ν = µ ◦ ν ◦ µ. Proof. Let S be regular. Let µ be a FGBI of S and ν be a FI of S. Then by Lemma 3.8, µ ◦ ν ◦ µ ⊆ µ ◦ χ ◦ µ ⊆ µ. Again, by Theorem 4.3[18], µ ◦ ν ◦ µ ⊆ χ ◦ ν ◦ χ ⊆ χ ◦ ν ⊆ ν. So µ ◦ ν ◦ µ ⊆ µ ∩ ν. Now let a ∈ S. Since S is regular there exist x ∈ S and α, 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)} = 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 FI of S) = (µ ∩ ν)(a). So µ ∩ ν ⊆ µ ◦ ν ◦ µ. Hence µ ◦ ν ◦ µ = µ ∩ ν. Conversely, let us suppose that the necessary condition holds. Let µ be a FGBI of S. Then by hypothesis, µ = µ ∩ χ = µ ◦ χ ◦ µ, where χ is the characteristic function of S. Hence by Theorem 3.10, S is regular. Theorem 3.12. Let S be a Γ-semigroup. Then following are equivalent: (1) S is regular, (2) A ∩ L ⊆ AΓL for each GBI, A of S and each LI, L of S, (3) R ∩ A ∩ L ⊆ RΓAΓL. 5
Proof. (1) ⇒ (2) : Let S be a regular Γ-semigroup and let a ∈ A ∩ L. Then there exist x ∈ S and α, β ∈ Γ such that a = aαxβa. Then a ∈ A and a ∈ L. Since L is a LI 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 LI and R is a RI of S, so xβa ∈ L and aαx ∈ R. This implies that a = aαxβaαxβa ∈ RΓAΓL. Hence R ∩ A ∩ L ⊆ RΓAΓL. (3) ⇒ (1) : 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 and α, β ∈ Γ such that a = aαxβa. Hence S is regular. (2) ⇒ (1) : Let A ∩ L ⊆ AΓL for each GBI A of S and for each LI 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. 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 3.13. Let S be a Γ-semigroup. then the following are equivalent: (1) S is regular, (2) µ ∩ ν ⊆ µ ◦ ν for each FBI, µ of S and for each FLI, ν of S, (3) µ ∩ ν ⊆ µ ◦ ν for each FGBI, µ of S and for each FLI, ν of S, (4) λ ∩ µ ∩ ν ⊆ λ ◦ µ ◦ ν for each FBI, µ of S, for each FLI, ν of S and for each FRI, λ of S, (5) λ ∩ µ ∩ ν ⊆ λ ◦ µ ◦ ν for each FGBI, µ of S and for each FLI, ν of S and for each FRI, λ of S. Proof. (1) ⇒ (2) : Let S be regular, µ be a FBI of S and ν be a FLI of S. Let a ∈ S.
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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 FBI of S and ν is a FLI of S) = (µ ∩ ν)(a). Hence µ ◦ ν ⊆ µ ∩ ν. (2) ⇒ (3) : Follows easily. (3) ⇒ (1) : Let us suppose that (3) holds. Let A be a GBI of S, L be a LI of S and a ∈ A ∩ L. Then a ∈ A and a ∈ L. Since A is a GBI of S, so by Theorem 3.3, χA is a FGBI of S, where χA is the characteristic function of A. By Theorem 3.1[18], χL is a FLI of S, where χL is the characteristic function of L. Hence by hypothesis χA ∩ χL ⊆ χA ◦ χL . Then (χA ◦ χL )(a) ≥ (χA ∩ χL )(a) = min{χA (a), χL (a)} = 1. Thus sup [min{χA (y), χL (z)}] = 1.
a=yγz
So there exist b, c ∈ S and δ ∈ Γ with a = bδc such that χA (b) = χL (c) = 1. Consequently, b ∈ A and c ∈ L. So a = bδc ∈ AΓL. So A ∩ L ⊆ AΓL. Hence by Theorem 3.12, S is regular. (1) ⇒ (4) : Let S is regular. Let µ be a FBI, ν be a FLI and λ be a FRI of S respectively. Let a ∈ S. Then there exist x ∈ S and 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 FRI 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)}](since µ is a FBI of S and ν is a FLI of S) ≥ min{λ(a), µ(a), ν(a)} = (λ ∩ µ ∩ ν)(a). Hence λ ∩ µ ∩ ν ⊆ λ ◦ µ ◦ ν.
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(4) ⇒ (5) : Follows easily. (5) ⇒ (1) : Let us suppose that (5) holds. Let A be a GBI of S, L be a LI of S, R be a RI of S and a ∈ R ∩ A ∩ L. Then a ∈ R, a ∈ A and a ∈ L. Since A is a GBI of S, so by Theorem 3.3, χA is a FGBI of S, where χA is the characteristic function of A. By Theorem 3.1[18], χL is a FLI of S and χR is a FRI of S where χL and χR are the characteristic functions of L and R respectively. Hence by hypothesis χR ∩ χA ∩ χL ⊆ χR ◦ χA ◦ χL . Then (χR ◦ χA ◦ χL )(a) ≥ (χR ∩ χA ∩ χL )(a) = min{χR (a), χA (a), χL (a)} = 1. Thus sup [min{(χR ◦ χA )(y), χL (z)}] = 1.
a=yγz
So there exist b, c ∈ S and δ ∈ Γ with a = bδc such that (χR ◦ χA )(b) = χL (c) = 1. Then c ∈ L and sup [min{χR (p), χA (q)}] = 1.
b=pρq
So there exist d, e ∈ S and θ ∈ Γ with b = dθe such that χR (d) = χA (e) = 1. Consequently, d ∈ R and e ∈ A. So a = bδc = dθeδc ∈ RΓAΓL. So R∩A∩L ⊆ RΓAΓL. Hence by Theorem 3.12, S is regular.
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