Commutative Orders Revisited

Victoria Gould, Pham Ngoc Anh, Pierre Grillet, Laszlo Marki

Research output: Contribution to journalArticlepeer-review


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
Original languageEnglish
Pages (from-to)336-366
Number of pages31
JournalSemigroup Forum
Issue number2
Early online date29 May 2014
Publication statusPublished - Oct 2014


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

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