Indecomposable, Projective and Flat S-Posets

For a monoid S, a (left) S-act is a non-empty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Right S-acts A can also be defined, and a tensor product A⊗S B (a set) can be defined that has the customary universal property with respect to balanced maps from A×B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (Walter de Gruyter, New York, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets (S-posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S-posets is given, based on strongly convex, indecomposable S-subposets, and a structure theorem for projective S-posets is given. A criterion for when two elements of the tensor product of S-posets is given, which is then applied to investigate several flatness properties.