Vol. 21, No. 1 (2015), pp. 19–23.
A SHORT NOTE ON BANDS OF GROUPS
B. Davvaze
1, and F. Sepahi
211Department of Mathematics, Yazd University, Yazd, Iran
2Department of Mathematics, Yazd University, Yazd, Iran
Abstract.In this paper, we give necessary and sufficient conditions on a semigroup Sto be a semilattice of groups, a normal band of groups and a rectangular band of groups.
Key words and Phrases: Semigroup, band, semilattice, band of semigroup.
Abstrak. Pada paper ini, kami menyatakan syarat perlu dan cukup dari suatu semigrup S untuk menjadi semilatis dari grup, pita normal dari grup, dan pita persegipanjang dari grup.
Kata kunci: Semigrup, pita, semilatis, pita dari semigrup.
1. Introduction and Preliminaries
Before we present the basic definitions we give a short history of the subject. In [4], Clifford introduced bands of semigroups and determined their structure. In [3], Ciric and S. Bogdanovic studied sturdy bands of semigroups. Then, this concept is studied by many authors, for example see [6, 11]. In [7, 8, 9, 10], Lajos studied semilattices of groups. In [1], Bogdanovic presented a characterization of semilattices of groups using the notion of weakly commutative semigroup. The purpose of this paper is as stated in the abstract.
A semigroupS is agroup, if for everya, b∈S,a∈bS∩Sb. A semigroupS is a band, if for every a∈S,a2=a. A commutative band is called a semilattice.
LetSbe a semigroup. If there exists a band{Sα|α∈ C}of mutually disjoint
subsemigroupsSα such that
(1) S= S
α∈C Sα,
(2) for everyα, β∈ C,SαSβ⊆Sαβ,
2010 Mathematics Subject Classification: 20M10.
then we sayS is a band of semigroups of typeC.
A congruenceρof a semigroupS is asemilattice congruenceofS if the factor S/ρ is a semilattice. If there exists a congruence relationρon a semigroupS such thatS/ρis a semilattice and everyρ-class is a group, then we saySis asemilattice of groups.
2. Main Results
LetSbe a semigroup. Then,S1is “Swith an identity adjoined if necessary”;
ifSis not already a monoid, a new element is adjoined and defined to be an identity. For an elementaofS, the relevant ideals are: (1) Theprincipal left ideal generated bya: S1a={sa|s∈S1}, this is the same as{sa|s∈S} ∪ {a}; (2) Theprincipal right ideal generated bya: aS1={as|s∈S1}, this is the same as{as|s∈S}∪{a}.
Leta, b∈S. We use the following well known notations:
a|rb ⇔ b∈aS1 and a|lb ⇔ b∈S1a,
a|tb ⇔ a|rb, a|lb.
For elementsa, b∈S, Green’s relationsL,Rand Hare defined by aLb ⇔ a|lb, b|la,
aRb ⇔ a|rb, b|ra,
aHb ⇔ a|tb, b|ta.
Indeed,H=L ∩ R.
Lemma 2.1. Ris a left congruence relation andL is a right congruence relation on S.
Proof. It is well-known in algebraic semigroup theory [4].
An elementx of a semigroup S is said to be left (right)regular ifx =yx2
(x =x2y) for some y ∈ S, or equivalently, xLx2 (xRx2). The second condition
in the following theorem is equivalent to a semigroup being left regular and right regular.
Theorem 2.2. A semigroup S is a semilattice of groups if and only if
(∀a, b∈S)ba|tab, a2|ta. (1)
Proof. Suppose that a semigroupS is a semilattice of groups andS= S
α∈C Sα. If
a∈Sα andb∈Sβ, thenab, ba∈Sαβ. SinceSαβ is a group,ba∈abS∩Sab. Since
a, a2∈S
α, we conclude thata2|ta.
Conversely, we define the relationη onS as follows:
a η b⇔a|tb, b|ta. (2)
c ∈S. Then, ac∈ bSc. Thus, there existst ∈ S such that ac =btc. By (1), we have
ac=btc∈btc2S⊆bc2tS⊆bcS.
Similarly, bc ∈ acS. Hence, acRbc and so R is a right congruence relation. By Lemma 2.1, we conclude thatRis a congruence relation. Sincea∈Sb, there exists m∈S such thata=mb. By (1), we obtain
ca=cmb∈Smcb⊆Scb.
So, L is a left congruence relation. By Lemma 2.1, we conclude that L is a con-gruence relation. Therefore, H=R ∩ Lis a congruence relation. For everya∈S, we have a2 ∈ aS∩Sa. Then, by (1), a ∈ a2S∩Sa2 which implies that aHa2.
Also, by (1), we obtainabHba. Therefore,His a congruence semilattice. Now, let S= S
Therefore,S is a semilattice of groupsSα.
Definition 2.3. A bandB is called normal if for everya, b, c∈ B,cabc=cbac.
Theorem 2.4. A semigroup S is a normal band of groups if and only if
(∀a, b, c, d∈S) abcd|tacbd, a|ta2. (3)
Proof. Suppose that a semigroupS is a normal band of groups andS= S
α∈C Sα.
Ifa∈Sα,b∈Sβ,c∈Sγ andd∈Sδ, thenabcd∈Sαβγδ. SinceCis a normal band,
acbdbacd,abdcacbd∈Sαβγδ. So, we have
(∀a, b, c, d∈S)abcd∈acbdbacdS⊆acbdS andabcd∈Sabdcacbd⊆Sacbd.
such that a=xb2. If α, β ∈ C, a, b∈S
α and x∈Sβ, then α=βα. By (3), for
every a∈S there existsy ∈S such that a=ya2. Ify ∈S
γ, thenα=γα. Thus,
we have
a=ya2=yaa=yaxb2=yaxbb∈Sγαβαb=Sαb.
Similarly, we can prove that b ∈Sαa. Since aRb, we conclude that a∈bSα and
b∈aSα. Therefore,Sα is a group andS is a normal band of groups.
Definition 2.5. A semigroup S is called a rectangular band if for everya, b∈S, aba=a.
Theorem 2.6. A semigroup S is a rectangular band of groups if and only if
(∀a, b∈S)a|taba. (4)
Proof.S Suppose that a semigroup S is a rectangular band of groups and S =
α∈C
Sα. Then, for everya, b∈S,aba∈S. Therefore,
a∈abaS anda∈Saba.
Conversely, suppose that (4) holds. If aHb, then for every c ∈ S we have ac ∈bSc ⊆bcbSc⊆bcS. Similarly, bc∈ acS and so acRbc. On the other hand, ca ∈ cSb ⊆ cSbcb ⊆ Scb and cb ∈ Sca. Thus, caLcb. Therefore, R is a right congruence relation and Lis a left congruence relation, and so His a congruence relation. Since for every a, b ∈ S, a ∈ Saba and a ∈ abaS, S is a congruence rectangular band.
Now, suppose that aHb. Then, aHb2 and there exists α ∈ C such that a, b ∈ Sα. So, there exist m, n ∈ S such that a= mb2 and b =na. If β, γ ∈ C,
m∈Sγ andn∈Sβ, thenα=γαandα=βα. So, we have
a=mb2=mnab∈Sγβαb=Sαb.
Similarly, we can prove thata∈bSαandb∈aSα∩Sαa. Therefore,Sαis a group.
Corollary 2.7. S is a left zero band of groups if and only if for every a, b ∈ S, a|tab.
3. Concluding Remarks
In this article, we studied some aspects of band of semigroups and groups. Let H be a non-empty set and let P∗(H) be the family of all non-empty subsets ofH. Ahyperoperation onH is a map⋆:H×H → P∗(H) and the couple (H, ⋆) is called ahypergroupoid. IfAandB are non-empty subsets ofH, then we denote A ⋆ B =Sa∈A, b∈Ba ⋆ b. A hypergroupoid (H, ⋆) is called asemihypergroupif for
Acknowledgement. The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper.
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