COMMUTATIVE ORDERS

A subsemigroup S of a semigroup Q is a left (right) order in Q if ever y q is an element of Q can be written as q = ab(q = ba*) for some a, b is an element of S, where a denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies i n a subgroup of Q. If S is both a left order and a right order in Q, w e 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 homomorphi c image of another. This enables us to show that certain semigroups ha ve maximum and minimum semigroups of quotients. We give examples to sh ow that this is not true in general.