a b s t r a c tIn this paper we study Morita contexts for semigroups. We prove a Rees matrix cover connection between strongly Morita equivalent semigroups and investigate how the existence of a unitary Morita semigroup over a given semigroup is related to the existence of a 'good' Rees matrix cover of this semigroup.
Abstract. In analogy to the xst-rings studied by García and Marín, we define fair semigroups and investigate Morita equivalence for a subclass of them. In particular, we present examples for semigroups which are Morita equivalent but not strongly Morita equivalent.
In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.
In 1971, inspired by the work of Lazard and Govorov for modules over a ring, Stenström proved that the strongly flat right acts AS over a monoid S (that is, the acts that are directed colimits of finitely generated free acts) are those for which the functor AS ⊗ − (from the category of left S-acts into the category of sets) preserves pullbacks and equalizers. He also provided interpolation-type conditions (now referred to in the literature as Property (P) and Property (E)) characterizing strong flatness. Unlike the situation for modules over a ring, strong flatness is strictly stronger than (mono-) flatness (wherein the functor AS ⊗ − is required only to preserve monomorphisms). The study of flatness properties of partially ordered monoids acting on partially ordered sets was initiated by S. M. Fakhruddin in the 1980s, and has been continued recently in the paper "Indecomposable, projective, and flat S-posets" by Shi, Liu, Wang, and Bulman-Fleming, Comm. Algebra 33, 235-251 (2005). In that paper, a criterion for the equality of elements in a tensor product of S-posets is given and a version of Property (P) is presented that, as in the unordered case, implies flatness and is implied by projectivity. The present paper introduces a corresponding Property (E) and establishes an analogue of the Lazard-Govorov-Stenström theorem in the context of S-posets.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.