Two classes of finite semigroups and monoids involving Lucas numbers.
type: Item
instance of: scholarly article
title: Two classes of finite semigroups and monoids involving Lucas numbers.
zbMATH Open document ID: 1190.20044
published in: Semigroup Forum
publication date: 2009-04-27T00:00:00Z
review text: As is depicted in this paper, the class of finitely presented groups \(\pi_1=\langle a,b;\;a^n=b^n,\;aba^{\tfrac n2}b^{\tfrac n2}=1\rangle\) is an extension of the class of triangle groups, and also their orders are finite depending on Lucas numbers. Generally, in this paper, by considering three presentations \(\pi_1\), \(\pi_2=\langle a,b;\;a^n=b^n,\;a^2ba^{\tfrac n2}b^{\tfrac n2}=a\rangle\) and \(\pi_3=\langle a,b;\;a^n=b^n,\;a^2ba^{\tfrac n2}b^{\tfrac n2+1}=ab\rangle\), it are studied the monoid \(\text{Mon}(\pi_i)\) for \(i=1,2,3\) and the semigroup \(\text{Sem}(\pi_i)\) for \(i=2,3\), and their relationship with the group \(\text{Grp}(\pi_i)\) for \(i=1,2,3\). In detail, it is proved in this paper that 1) For all \(n\geq 2\), \(\text{Mon}(\pi_1)\) is actually a group and \(\text{Mon}(\pi_1)\cong\text{Grp}(\pi_1)\). 2) For all \(n\geq 2\), \(|\text{Sem}(\pi_2)|=|\text{Grp}(\pi_2)|+n-1\). 3) For all \(n\geq 2\), \(|\text{Sem}(\pi_3)|=|\text{Grp}(\pi_3)|+n^2\).
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 5547497
zbMATH Keywords: finite semigroups
zbMATH Keywords: finite monoids
zbMATH Keywords: presentations
zbMATH Keywords: Lucas numbers
zbMATH Keywords: finitely presented groups
zbMATH Keywords: finitely presented semigroups
zbMATH Keywords: finitely presented monoids
author: Karim Ahmadidelir
author: C. M. Campbell
author: Hossein Doostie
reviewed by: A. Sinan Çevik
MaRDI profile type: Publication
full work available at URL: s00233-008-9119-8
OpenAlex ID: W2084685707
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DOI: 10.1007/S00233-008-9119-8
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description: scientific article; zbMATH DE number 5547497