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Semigroup and Group Presentations

Semigroup and Group Presentations C. M. CAMPBELL, E. F. ROBERTSON, N. RUSKUC AND R. M. THOMAS 1. Introduction We consider semigroup presentations of the form n = <fl ...,a \u = v ,...,u = v ) ls n l l m m where m,neN, and u ,v , i = 1,...,m, are non-empty words in the symbols a,...,a. t t x n Every such presentation can also be considered as a group presentation. To distinguish between the semigroup and the group defined by II, we shall denote them by Sgp(II) and Gp(II) respectively. The purpose of this paper is to explore the relationship between Sgp(Il) and Gp(II). Examples of semigroups defined by presentations can be found in [1], [2], [7] and [9]. A common feature of all these semigroups is that each of them contains a certain number of copies of the group with the same presentation as the semigroup. On the other hand, the presentation <a| > shows not only that Gp(IT) is not necessarily contained in Sgp(IT), but that it can even fail to be a homomorphic image of Sgp(II). Although it is fairly easy to show that if either Gp(FI) or Sgp(II) is finite, then Gp(II) is a homomorphic image http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Semigroup and Group Presentations

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/27.1.46
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Abstract

C. M. CAMPBELL, E. F. ROBERTSON, N. RUSKUC AND R. M. THOMAS 1. Introduction We consider semigroup presentations of the form n = <fl ...,a \u = v ,...,u = v ) ls n l l m m where m,neN, and u ,v , i = 1,...,m, are non-empty words in the symbols a,...,a. t t x n Every such presentation can also be considered as a group presentation. To distinguish between the semigroup and the group defined by II, we shall denote them by Sgp(II) and Gp(II) respectively. The purpose of this paper is to explore the relationship between Sgp(Il) and Gp(II). Examples of semigroups defined by presentations can be found in [1], [2], [7] and [9]. A common feature of all these semigroups is that each of them contains a certain number of copies of the group with the same presentation as the semigroup. On the other hand, the presentation <a| > shows not only that Gp(IT) is not necessarily contained in Sgp(IT), but that it can even fail to be a homomorphic image of Sgp(II). Although it is fairly easy to show that if either Gp(FI) or Sgp(II) is finite, then Gp(II) is a homomorphic image

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1995

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