Commutative orders revisited

Abstract

This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order $S$, the square-cancellable elements $\mathcal {S}(S)$ constitute a well-behaved separable subsemigroup. Indeed, $\mathcal {S}(S)$ is also an order and has a maximum semigroup of quotients $R$, which is Clifford. We present a new characterisation of commutative orders in terms of semilattice decompositions of $\mathcal {S}(S)$ and families of ideals of $S$. We investigate the role of tensor products in constructing quotients, and show that all semigroups of quotients of $S$ are homomorphic images of the tensor product $R\otimes _{\mathcal {S}(S)} S$. By introducing the notions of generalised order and semigroup of generalised quotients, we show that if $S$ has a semigroup of generalised quotients, then it has a greatest one. For this we determine those semilattice congruences on $\mathcal {S}(S)$ that are restrictions of congruences on $S$.