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Commutative orders

Published online by Cambridge University Press:  14 November 2011

David Easdown  and
Victoria Gould
David Easdown
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia
Victoria Gould
Affiliation:
Department of Mathematics, University of York, Heslington, York YOl 5DD, U.K.
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Abstract

A subsemigroup S of a semigroup Q is a left (right) order in Q if every q ∈ Q can be written as q = a*b(q = ba*) for some a, b ∈S, where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. If S is both a left order and a right order in Q, we say that S is an order in Q. We show that if S is a left order in Q and S satisfies a permutation identity xl…xn = x1π…xnπ where 1 < 1π and nπ<n, then S and Q are commutative. We give a characterisation of commutative orders and decide the question of when one semigroup of quotients of a commutative semigroup is a homomorphic image of another. This enables us to show that certain semigroups have maximum and minimum semigroups of quotients. We give examples to show that this is not true in general.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1201 - 1216
Copyright
Copyright © Royal Society of Edinburgh 1996

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