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
davvaz@yazd.ac.ir
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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