Commutative Orders


David Easdown and Victoria Gould


Research Report 96-4
To appear in Proceedings Royal Society Edinburgh
Date: December 1995


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


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