EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 1, 2014, 77-85 ISSN 1307-5543 – www.ejpam.com The Relation B and Minimal bi–ideals in Γ–semigroups Islam Braja1, Petraq Petro2,∗ 1 Department of Mathematics, Faculty of Natural Sciences, University “A. Xhuvani”, Elbasan, Alba- nia 2 Department of Mathematics, Faculty of Natural Sciences, University of Tirana, Tirana, Albania Abstract. In this paper we introduce the relation B “to generate the same principal bi-ideal” in Γ– semigroups. One of the main results that are proved here is the analogue of the Green’s Theorem for Γ–semigroups, which we call the Green’s Theorem for the relation B in Γ–semigroups. Applying our Green’s Theorem for relationB in Γ–semigroups, we prove that any bi-ideal of a Γ–semigroup without zero is minimal if and only if it is a Γ–subgroup. Further, we prove that, if a Γ–semigroup M without zero has a cancellable element contained in a minimal bi-ideal B of M , then M is a Γ–group. Finally, we prove that, if for elements a, c of a Γ–semigroup without zero we have aDc and the principal bi-ideal (a)b and principal quasi-ideal (a)q are minimal, then (a)b = (a)q and the principal bi-ideal (c)b and the principal quasi-ideal (c)q are minimal too, and (c)b = (c)q. Key Words and Phrases: Γ–semigroup, Green’s theorem, quasi–ideal, bi–ideal, Γ–group. 1. Introduction The notion of Γ–semigroup is introduced by Sen in [8]. Let M and Γ be non–empty sets. Any map from M ×Γ×M to M will be called a Γ–multiplication in M and is denoted by (·)Γ. The result of this Γ–multiplication for a, b ∈ M and γ ∈ Γ is denoted by aγb. According to Sen and Saha [9], a Γ–semigroup is an ordered pair (M , (·)Γ), where M and Γ are non-empty sets and (·)Γ is a Γ–multiplication in M for which the following proposition: ∀(a, b, c,α,β) ∈ M3×Γ2, (aαb)β c = aα(bβ c) is true. In the literature there are many examples of Γ–semigroups, but the following example, which is inspired from Hestenes’s rings [3] is the most well known one. ∗Corresponding author. Email addresses: braja_islam@yahoo.com (I. Braja), petropetraq@yahoo.com (P. Petro) http://www.ejpam.com 77 c© 2014 EJPAM All rights reserved. I. Braja, P. Petro / Eur. J. Pure Appl. Math, 7 (2014), 77-85 78 Example 1. The Γ–semigroup M of all m×n matrices with entries from a field F, where Γ is the set of all n×m matrices, with entries from F. The result of Γ–multiplication in M for two m× n matrices A, B and an n×m matrix C is the usual product ACB. Note that every plain semigroup S can be considered as a Γ–semigroup by taking as Γ a singelton {1}, where 1 is the identity element of S, when S has a such element, or it is a symbol not representing an element of S, and the Γ–multiplication in S is defined by a1b = ab, where ab is the usual product in plain semigroup S. Similarly to the definition of relationsRplain,Lplain,Hplain and Dplain in plain semigroup, Saha in [7] has introduced the analogue relations R , L , H , D in a Γ–semigroup M , which are called the Green’s relations in the Γ–semigroup M . In this paper we define the relation B in a Γ–semigroup M such that aB c if and only if (a)b = (c)b, where (a)b and (c)b are the principal bi–ideals generated by elements a, c of M respectively. The definition of relation B in Γ-semigroups mimics the definition of relation B in plain semigroup introduced in [4]. We show that in Γ–semigroups the relation B is different from Green’s relation H . One of our main results claims that the analogue of Green’s Theorem for Green’s relation Hplain =Rplain ∩Lplain holds true for the relationB in Γ -semigroups. This theorem we call Green’s Theorem for the relationB in Γ–semigroups. From this theorem, as a particular case, we get a Green’s Theorem for the relation B in plain semigroups. Then we use our theorem for the relation B in Γ–semigroups to prove that any bi-ideal of a Γ–semigroup without zero is minimal if and only if it is a Γ–subgroup. As a corollary of the above result we get the analogue of the result for minimal quasi-ideal in Γ–semigroup [6], which states that “if a Γ–semigroup M without zero has a cancellable element contained in a minimal bi-ideal B of M , then M is a Γ–group”. From this result we get the analogue of result for plain semigroups which state that “if a semigroup S without zero has a cancellable element contained in a minimal bi–ideal, then S is group”. At last, we show that if for the elements a, c of Γ–semigroup M without zero, we have aDc and principal bi-ideal (a)b and principal quasi–ideal (a)q are minimal, then (a)b = (a)q and the principal bi-ideal (c)b and the principal quasi-ideal (c)q are minimal too, and (c)b = (c)q. At the end of this paper we raise an open problem. 2. Preliminaries We give some notions and present some auxiliary results that will be used throughout the paper. Let M be a Γ–semigroup and A, B be subsets of M . We define the set AΓB = {aγb ∈ M |a ∈ A, b ∈ B and γ ∈ Γ}. For simplicity we write aΓB instead of {a}ΓB, AΓb in place of AΓ{b}, and aΓb instead of {a}Γ{b}. Analogously with the definitions in plain semigroups there are given the following defini- tions in Γ–semigroups. I. Braja, P. Petro / Eur. J. Pure Appl. Math, 7 (2014), 77-85 79 Definition 1. Let M be a Γ–semigroup. A non–empty subset M1 of M is said to be a Γ– subsemigroup of M if M1ΓM1 ⊆ M1. Definition 2. A right [left] ideal of a Γ–semigroup M is a non–empty subset R [L] of M such that RΓM ⊆ R, [MΓL ⊆ L]. Definition 3. A quasi–ideal of a Γ–semigroup M is a non–empty subset Q of M such that QΓM ∩MΓQ ⊆Q. Definition 4. A bi–ideal of a Γ–semigroup M is a Γ–subsemigroup B of M such that BΓMΓB ⊆ B. Similarly to the plain semigroups, it is easy to prove the following two propositions: Proposition 1. Every quasi–ideal of a Γ–semigroup M is a bi–ideal of M. Proposition 2. The intersection of any set of bi-ideals of a Γ–semigroup M is an empty set or is a bi-ideal of M. Theorem 1 ([1]). Let A be a nonempty subset of a Γ–semigroup M. Then (A)b = A∪ AΓA∪ AΓMΓA, where (A)b is the smallest bi–ideal of M containing A, i.e. the intersection of bi–ideals of M containing A. Let M be a Γ–semigroup and γ ∈ Γ is a fixed element. As in [9], we define the multipli- cation ◦ in M by a ◦ b = aγb. It is obvious that ◦ is associative, hence we obtain a semigroup (M ,◦) which is shortly denoted by Mγ. A zero of a Γ–semigroup M is an element 0 of M such that for all a ∈ M and γ ∈ Γ we have aγ0= 0γa = 0. Theorem 2 ([6]). Let M be any Γ–semigroup without zero and γ ∈ Γ a fixed element. Then Sγ is a group if and only if S has not proper quasi-ideals. From this theorem we give: Theorem 3 (citesensaha). Let M be a Γ–semigroup without zero. If Mγ is a group for some γ ∈ Γ, then it is a group for all γ ∈ Γ. Definition 5 ([9]). A Γ–semigroup M is called a Γ–group if Mγ is a group for some (hence for all) γ ∈ Γ. Let M be a Γ–subsemigroup of a Γ–semigroup M . In the set M we have a Γ–multiplication induced by the Γ–multiplication of Γ–semigroup M , (·)Γ, denoting it with the same symbol. It is clear that the ordered pair (M , (·)Γ) is a Γ–semigroup. From Theorem 3, if Mγ is a group for some γ ∈ Γ, then it is group for all γ ∈ Γ and so it is a Γ–group. In this case, we will call M a Γ–subgroup of the Γ–semigroup M . I. Braja, P. Petro / Eur. J. Pure Appl. Math, 7 (2014), 77-85 80 Saha has defined in [7] the Green’s relations R , L ,H in a Γ–semigroup M as follows: ∀(a, b) ∈M2, aR b⇔ (a)r = (b)r , ∀(a, b) ∈M2, aL b⇔ (a)l = (b)l , ∀(a, b) ∈M2, aH b⇔ (a)r = (b)r and (a)l = (b)l , where (a)r = a∪ aΓM , (b)r = b∪ bΓM , (a)l = a∪MΓa, (b)l = b∪MΓb, are respectively the principal right ideal generated by a, the principal right ideal generated by b, the principal left ideal generated by a, and the principal left ideal generated by b in Γ–semigroup M . It turns out that R , L , H are equivalent relations. The respective equivalence classes of a ∈ M are denoted by Ra, La, Ha. Proposition 3 ([7]). Let M be a Γ–semigroup. Then we have: (i) For every three elements a, b, c of M and for every γ ∈ Γ aR b⇒ cγaRcγb and aL b⇒ aγcL aγb. (ii) For every two elements a, b ∈ M, aR b if and only if either a = b or there exist α,β ∈ Γ and c, d ∈ M such that a = bαc and b = αβd. (iii) For every two elements a, b ∈ M, aL b if and only if either a = b or there exist α,β ∈ Γ and c, d ∈ M such that a = cαb and b = dβα. (iv) R and L commute, that is R ◦L = L ◦R . So, one can define a fourth Green’s relation in M which is D =R ◦L =L ◦R . The equivalence class of a ∈ M from D is denoted by Da. In [6], it is defined the relation Q in a Γ–semigroup M as follows ∀(a, b) ∈ M2, aQb⇔ (a)q = (b)q, where (a)q = a ∪ (aΓM ∩ MΓa), (b)q = b ∪ (bΓM ∩ MΓb), are the principal quasi-ideal generated by a and the principal quasi-ideal generated by b in Γ –semigroup M , respectively. The relation Q is an equivalence relation and moreover we have Proposition 4 ([6]). The relationsH and Q coincide in every Γ–semigroup M. For the relationH , consequently for relation Q, has an analogue of Green’s Theorem for plain semigroups, which is called Green’s Theorem for Γ–semigroups. Theorem 4 ([6] Green’s Theorem for Γ–semigroups). If the elements a, b, aγb of a Γ–semigroup M all belong to the sameH –class H of M, then H is a subgroup of the semigroup Mγ. Moreover, for any two element h1, h2 ∈ H, the element h1γh2 belongs to H. I. Braja, P. Petro / Eur. J. Pure Appl. Math, 7 (2014), 77-85 81 Theorem 5 ([6]). A quasi–ideal Q of a Γ–semigroup S without zero is minimal if and only if Q is anH –class. Theorem 6 ([6]). A quasi–ideal Q of a Γ–semigroup S without zero is minimal if and only if Q is a Γ–subgroup of S. Theorem 7 ([6]). Let a, b be two elements of a Γ–semigroup S without zero such that aDb. Then the principal quasi–ideal (a)q is minimal if and only if the same holds for (b)q. 3. Main Results For every element a of a Γ–semigroup M we denote by (a)b the intersection of all bi-ideals of M that contain a. This bi–ideal, that is, the smallest bi–ideal of M containing a, is called principal bi-ideal of M generated by a. From the Theorem 1 we have (a)b = a ∪ aΓa ∪ aΓMΓa. Now, similarly with the definition of the relation B in plain semigroups [4], we define the relationB in Γ–semigroup M by ∀(a, c) ∈ M2, aB c⇔ (a)b = (c)b. So, for every two elements a, c of Γ–semigroups we have aB c⇔ a ∪ aΓa ∪ aΓMΓa = c ∪ cΓc ∪ cΓMΓc. Clearly B is an equivalence relation on M . The equivalence class of M modB containing the element a ∈ M is denoted by Ba. From Proposition 3, it is clear that B ⊆H . The following example shows that the inclu- sion may be strict. Example 2. Consider the set of integers modulo 8, Z/8Z= {0, 1, 2, 3, 4, 5, 6, 7}, and Γ = {0,1, 2} ⊆ N∪ {0}. The result of Γ–multiplication in M = Z/8Z for two any elements a, b of M and every element γ ∈ Γ is the usual product aγb of integers modulo 8, a,γ, b . It is clear that (M = Z/8Z, (·)Γ) is a Γ–semigroup. The elements 2 and 6 are L equivalent since: (2)l =2∪Z/8ZΓ2= {0, 2, 4, 6}, (6)l =6∪Z/8ZΓ6= {0, 2, 4, 6}. In the Γ–semigroup (M = Z/8Z, (·)Γ) the Green’s relations L and R coincide and so we have 2H 6. I. Braja, P. Petro / Eur. J. Pure Appl. Math, 7 (2014), 77-85 82 The elements 2, 6 of Γ–semigroup M = Z/8Z are notB–equivalent since: (2)b =2∪ 2Γ2∪ 2ΓMΓ2= {0, 2, 4}, (6)b =6∪ 6Γ6∪ 6ΓMΓ6= {0, 4, 6}, and soH 6=B . For the equivalence relation B in Γ–semigroup it is true the following theorem, which resembles the Green’s Theorem for plain semigroups [10], the Green’s Theorem for rings [5], the Green’s Theorem for semirings [2], and Green’s Theorem for Γ–semigroup [6]. We will call this theorem the Green’s Theorem for the relationB in Γ–semigroups. Theorem 8. If the elements a, b, aγb of a Γ–semigroup (M , (·)Γ) all belong to the sameB-class B, then B is a Γ–subgroup of semigroup Mγ. Proof. Since the relationB is included in the relationH , we have B = Ba ⊆ Ha. Thus the elements a, b, aγb belong to theH –class Ha of Γ –semigroup M . So, by Theorem 3, Ha is a subgroup of semigroup Mγ and therefore there exists the identity e of subgroup Ha and the following equalities are true: a = eγaγe, e = aγa−1γa−1γa, where a−1 is the inverse element of a in the subgroup Ha of semigroup Mγ. These equalities show that the principal bi–ideals generated by elements a and e are the same. Thus, the element e belongs to the class Ba = B. Now, let x be any element of subgroup Ha. We have the following equalities: x = eγxγe, e = xγx−1γx−1γx , where x−1 is the inverse element of the element x of subgroup Ha of semigroup Mγ. These equalities show that (x)b = (e)b. So, since e ∈ Ba, the element x belongs to Ba. Thus, we have B = Ba = Ha, and consequentlyB-class B is a subgroup of Γ–semigroup Mγ. A element e of a Γ-semigroup M is called idempotent if there exists γ ∈ Γ such that e = eγe. From the Theorem 8 we get immediately the following: Corollary 1. If aB–class B of a Γ–semigroup M contains an idempotent e = eγe,γ ∈ Γ, then B is a subgroup of semigroup Mγ. Since, every plain semigroup S can be considered as a Γ–semigroup, from the Theorem 8 and the Corollary 1, we get the following theorem and corollary to plain semigroups: Theorem 9. If the elements a, b, ab of a semigroup S all belong to the same B–class B, then B is a subgroup of semigroup S. I. Braja, P. Petro / Eur. J. Pure Appl. Math, 7 (2014), 77-85 83 Corollary 2. If an B–class B of a semigroup S contains an idempotent e, then B is a subgroup of semigroup S. We can call the Theorem 9 the Green’s Theorem for the relationB in plain semigroups. Proposition 5. Let M be an arbitrary Γ–semigroup. If the idempotent e = eγe,γ ∈ Γ together with a, b ∈ M all belong to the sameB-class B, then eγa = aγe = a and aγb ∈ B. Proof. Since B contains an idempotent e = eγe,γ ∈ Γ, then the Corollary 1 implies that B is a subgroup of Mγ. The identity of B is e because e ◦ e = eγe = e. Since a ◦ e = e ◦ a = e, we have aγe = a = aγe. The element a, b belongs to B, therefore aγb = a ◦ b ∈ B. A bi–ideal B of a Γ–semigroup M without zero is called minimal if B does not properly contain any bi–ideal of M . One can prove easily that: Lemma 1. A bi–ideal B of a Γ–semigroup M without zero is minimal if and only if B is an B–class. Now we will use the Green’s Theorem for the relation B in Γ–semigroup (Theorem 8) to prove a theorem concerning minimal bi–ideals in Γ–semigroup M without zero. Theorem 10. A bi–ideal B of a Γ–semigroup M without zero is minimal if and only if B is a Γ–subgroup of M. Proof. If B is a minimal bi–ideal of the Γ–semigroup M , then by Lemma 1 all elements of B are B–equivalent. Thus for two elements a, b of B and every γ ∈ Γ the elements a, b, aγb all belong to the same B–class of M . Now applying the Green’s Theorem for the relation B in Γ–semigroup (Theorem 8) the B–class B is a subgroup of semigroup Mγ for every γ ∈ Γ. So, B is a Γ–subsemigroup such that for every γ ∈ Γ, Bγ = (B,◦) is a group. Thus B is a Γ–subgroup of Γ–semigroup M and it is aH –class. Conversely, let the bi–ideal B be a Γ–subgroup of M . If B′ is a bi–ideal of M contain in B, then B′ΓBΓB′ ⊆ B′ΓMΓB′ ⊆ B′, that is, B′ is a bi–ideal of B, too. Let a be an element of B′ and γ an element of Γ. Then, since the semigroup Bγ = (B,◦) is a group, we have B = a ◦ B ◦ a = aγBγa ⊆ B′, whence B = B′. This means that B is a minimal bi–ideal of Γ–semigroup M . As a particular case we get the following theorem for plain semigroups: Theorem 11. A bi–ideal B of a plain semigroup S without zero is minimal if and only if B is a subgroup of S. This theorem is proved in [4] by a direct method. I. Braja, P. Petro / Eur. J. Pure Appl. Math, 7 (2014), 77-85 84 Definition 6. An element a of a Γ–semigroup M is called cancellable if for two elements b, c ∈ M and every γ ∈ Γ we have (aγb = aγc⇒ b = c)∧ (bγa = cγa⇒ b = c). Theorem 12. If a Γ–semigroup M without zero has a cancellable element contained in a minimal bi–ideal B of M, then M is a Γ–group. Proof. By the Theorem 10, the minimal bi–ideal B is a Γ–subgroup of M . Let e be the identity of the group Bγ = (B,◦) for a fixed γ ∈ Γ and let a be a cancellable element of M contained in B. Then multiplying both side of the equality eγa = a by any element b of M , we have bγeγa = bγa, hence bγe = b. Dually we obtain eγb = b for every b ∈ M . Thus e is the identity element of the semigroup Mγ = (M ,◦). Since e ∈ B, for any b ∈ M we have b = eγbγe ∈ B. So, M = B and consequently M is a Γ–group with zero. Since every plain groups is a minimal bi–ideal and has a cancellable element (this is the identity element of the group), therefore from the Theorem 12, we get the following: Theorem 13. A semigroup S without zero is a group if and only if it has a cancellable element contained in a minimal bi–ideal B of S. Theorem 14. Let a, c are two elements of a Γ–semigroup without zero such that aDc. If the principal bi–ideal (a)b and the principal quasi–ideal (a)q are minimal, then (a)b = (a)q and the principal bi–ideal (c)b and the principal quasi–ideal (c)q are minimal and (c)b = (c)q. Proof. Assume that bi–ideal (a)b and quasi–ideal (a)q are minimal. Firstly we prove that (a)b = (a)q. It is clear the inclusion (a)b ⊆ (a)q. Since (a)q is a minimal quasi–ideal, then Theorem 5 implies that (a)q = Ha. So, we have Ba = (a)b ⊆ (a)q ⊆ Ha. By Theorem 4, Ha is a Γ–subgroup, therefore Ha ⊆ Ba and consequently (a)b = (a)q. Since the principal quasi–ideal (a)q is minimal, then the Theorem 7 implies that the prin- cipal quasi–ideal (c)q is minimal. Now from the Theorem 5 we have Bc ⊆ (c)b ⊆ (c)q = Hc . By the Theorem 4, Hc is a Γ–group, therefore Hc ⊆ Bc and consequently there are true the equalities Bc = (c)b = (c)q = Hc . So, (c)b = (c)q. At the end of this paper, we raise the following open problem: Problem. 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