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Some classes of finite semigroups with kite-like egg-boxes of D-classes

Published online by Cambridge University Press:  05 July 2011

K. Ahmadidelir
Affiliation:
Islamic Azad University — Tabriz Branch, Iran
H. Doostie
Affiliation:
Tarbiat Moallem University, Iran
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

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 Gpi) has been found as follows:

Theorem 1.1For every n ≥ 2, |Sg2)| = |Gp2)| + n - 1.

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Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] K., Ahmadidelir, C.M., Campbell and H., Doostie, Almost commutative semigroups, Algebra Colloquium, Accepted: 29 December 2008.Google Scholar
[2] K., Ahmadidelir, C.M., Campbell and H., Doostie, Two classes of finite semigroups and monoids involving Lucas numbers, Semigroup Forum 78 (2009), 200–209.Google Scholar
[3] K., Ahmadidelir, H., Doostie and R., Gholami, On the structure of generalized polyhedral semigroups with zero deficiency, Submitted to Nagoya Math. Journal.
[4] H., Doostie and K., Ahmadidelir, A class of Z-metacyclic groups involving the Lucas numbers, Novi-Sad J. Math., Accepted: 10 May 2009.Google Scholar
[5] C.M., Campbell, E.F., Robertson, N., Ruskuc and R.M., Thomas, Semigroup and group presentations, Bull. London Math. Soc. 27 (1995), 46–50.Google Scholar
[6] C.M., Campbell, J.D., Mitchell and N., Ruskuc, Comparing semigroup and monoid presentations for finite monoids, Monatsh. Math. 134 (2002), 287–293.Google Scholar
[7] C.M., Campbell, J.D., Mitchell and N., Ruskuc, On defining groups efficiently without using inverses, Math. Proc. Cambridge Philos. Soc. 133 (2002), 31–36.Google Scholar
[8] E.F., Robertson and Y., Ünlü, On semigroup presentations, Proc. Edinburgh Math. Soc. 36 (1993), 55–68.Google Scholar

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