Lazard's Theorem forS‐posets

Abstract

AbstractIn 1971, inspired by the work of Lazard and Govorov for modules over a ring, Stenström proved that the strongly flat right actsASover a monoidS(that is, the acts that are directed colimits of finitely generated free acts) are those for which the functorAS⊗ (from the category of leftS‐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 functorAS⊗ 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 flatS‐posets” by Shi, Liu, Wang, and Bulman–Fleming, Comm. Algebra33, 235–251 (2005). In that paper, a criterion for the equality of elements in a tensor product ofS‐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 ofS‐posets. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)