Flatness Properties of S-Posets

ABSTRACT Let S be a partially ordered monoid, or briefly, pomonoid. A right S-poset (often denoted A S ) is a poset A together with a right S-action (a,s)↝ as that is monotone in both arguments and that satisfies the conditions a(st) = (as)t and a1 = 1 for all a ∊ A, s,t ∊ S. Left S-posets S B are defined analogously, and the left or right S-posets form categories, S-POS and POS-S, whose morphisms are the monotone maps that preserve the S-action. In these categories, as in the category POS of posets, the monomorphisms and epimorphisms are the injective and surjective morphisms, respectively, but the embeddings and quotient maps have stronger properties; in particular, an embedding is a monomorphism that is also an order embedding. A tensor product A S ⊗  S B exists (a poset) that has the customary universal property with respect to balanced, bi-monotone maps from A  ×  B into posets. Various flatness properties of A S can be defined in terms of the functor A S ⊗ − from S-POS into POS. More specifically, a...