Band congruences in ordered semigroups
Niovi Kehayopulu, Michael Tsingelis · International Mathematical Forum · 2007
The semilattice congruences play an important role in studying the decomposition of semigroups-without order. When we pass from semi-groups without order to ordered semigroups, the same role is played by the complete semilattice congruences. So, the complete semilattice con-gruences play an essential role in studying the structure, especially the decomposition of ordered semigroups. The characterization of complete semilattices of semigroups of a given type has been already considered by the same authors. Band congruences have been also proved to be use-ful in studying the decomposition of some types of ordered semigroups, especially the decomposition of t-archimedean ordered semigroups. In this paper we prove that an ordered semigroup S is a band of semi-groups of a given type T if and only if it is decomposable into pairwise disjoint subsemigroups Sα of S of type T indexed by a band B, such that SαSβ ⊆ Sαβ for all α, β ∈ B and Sα ∩ (Sβ] = ∅ implies α = αβ = βα.