Commutative Orders Revisited

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Publication details

JournalSemigroup Forum
DateE-pub ahead of print - 29 May 2014
DatePublished (current) - Oct 2014
Issue number2
Number of pages31
Pages (from-to)336-366
Early online date29/05/14
Original languageEnglish


This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order S, the square-cancellable elements S(S) constitute a well-behaved separable subsemigroup. Indeed, 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 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⊗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 S(S) that are restrictions of congruences on S

    Research areas

  • Commutative semigroup, Tensor product, (Generalised) order, Semigroup of (generalised) quotients, Square-cancellable elements

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