# Commutative Orders

## Authors

**David Easdown** and **Victoria Gould**

## Status

Research Report 96-4

To appear in Proceedings Royal Society Edinburgh

Date: December 1995

## Abstract

A subsemigroup S of a semigroup Q is a *left (right) order*
in Q if every q in Q can be written as q=a*b (q=ba*) for some
a, b in 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
x_1...x_n=x_{1\pi}...x_{n\pi} where 1 < 1\pi and n\pi < 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.

## Key phrases

commutative semigroups of quotients.

## AMS Subject Classification (1991)

Primary: 20M10 20M14

Secondary:

## Content

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