R-bands of r-archimedean ordered semigroups

Abstract

Some semigroups (without order) are decomposable into r-archimedean semigroups. In the present paper we deal with the problem of decomposing certain ordered semigroups into r (l)-archimedean components. The semilattice congruences play an important role in the decomposition of semigroups -without order. When we pass from semigroups without order to ordered semigroups, the same role is played by the complete semilattice congruences. The characterization of complete semilattices of semigroups of a given type has been considered by the same authors. Band congruences play an important role in studying the decomposition of some ordered semigroups, like the decomposition of t-archimedean ordered semigroups. The r (l)-band congruences have been also proved to be useful in studying the decomposition of some types of ordered semigroups, especially the decomposition of r (l)-archimedean ordered semigroups. In this paper we first prove that an ordered semigroup S is an r (resp. l)-band of semigroups of a given type \( \mathcal{T} \) if and only if it is decomposable into pairwise disjoint subsemigroups Sα of S of type \( \mathcal{T} \) indexed by a band B such that SαSβ ⊆ Sαβ for all α, β ∈ B and Sα ∩ (Sβ] ≡ O implies α = βα (resp. α = αβ). Thenwe characterize the r (resp. l)-bands of r (resp. l)-archimedean semigroups.