Abstract

In this article, by considering some presentations of groups and semigroups, we investigate the structure of the groups and semigroups presented by them and introduce some infinite classes of semigroups which have kite-like egg-boxes of D-classes. These semigroups have a unique idempotent and the minimal two-sided ideal of them is isomorphic to the group presented by the same presentation as for these semigroups. All of the Green's relations in these semigroups coincide and every proper subsemigroup of them is a subgroup.

Introduction

Let π be a semigroup and/or group presentation. To avoid confusion we denote the semigroup presented by π by Sg(π) and a group presented by π by Gp(π).

The class of deficiency zero groups presented by

has been studied in [4] where the corresponding group has been proved to be finite of order

for every integer n ≥ 2, where ⌊t⌋ denotes the integer part of a real t and is the sequence of Lucas numbers

In [4], it has been proved that all of these groups are metabelian and that if n ≡ 0 (mod 4) or n ≡ ±1 (mod 6) they are metacyclic.

Also, for every integer n ≥ 2, the presentations

and

of semigroups have been studied in [2] by the authors of this article and in that investigation, their relationship with Gp(πi) has been found as follows:

Theorem 1.1For every n ≥ 2, |Sg(π2)| = |Gp(π2)| + n - 1.