Flatness Properties of S-Posets

Abstract

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, an S-poset A S is called flat if the induced morphism A S ⊗ S B → A S ⊗ S C is injective whenever S B → S C is an embedding in S-POS: this means that, for all S B and all a,a ′ ∈ A and b,b ′ ∈ B, if a⊗ b = a ′⊗ b ′ in A⊗ B, then the same equality holds in A⊗ (Sb ∪ Sb ′). A S is called (principally) weakly flat if the induced morphism above is injective for all embeddings of (principal) left ideals into S S. Similarly, A S is called po-flat if the functor A S ⊗ − preserves embeddings: for this definition, replace = by ≤ in the description above (see Shi, 2005). Weak and principally weak versions of po-flatness are defined in an obvious way. In the present article, we first consider flatness properties for one-element and Rees factor S-posets. We present examples that distinguish between various types of flatness and the corresponding, generally stronger, notions of po-flatness. We then initiate a study of absolute flatness for pomonoids: a monoid (respective pomonoid) S is called right absolutely flat if all right S-acts (respective S-posets) are flat. The findings for absolute flatness of pomonoids are markedly different from the corresponding unordered results.