A SHORT NOTE ON BANDS OF GROUPS

Bijan Davvaz

doi:10.22342/jims.21.1.151.19-23

Abstract

In this paper we give necessary and sufficient conditions on a semigroup S that it be semilattice of groups, a normal band of groups, and a matrix of groups.DOI : http://dx.doi.org/10.22342/jims.21.1.151.19-23

Highlights

Introduction and PreliminariesBefore we present the basic definitions we give a short history of the subject
In [1], Bogdanovic presented a characterization of semilattices of groups using the notion of weakly commutative semigroup
For an element a of S, the relevant ideals are: (1) The principal left ideal generated by a: S1a = {sa | s ∈ S1}, this is the same as {sa | s ∈ S} ∪ {a}; (2) The principal right ideal generated by a: aS1 = {as | s ∈ S1}, this is the same as {as | s ∈ S}∪{a}

Summary

Introduction

Before we present the basic definitions we give a short history of the subject. 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. A semigroup S is a group, if for every a, b ∈ S, a ∈ bS ∩ Sb. A semigroup S is a band, if for every a ∈ S, a2 = a. A commutative band is called a semilattice.

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Conclusion