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, 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.

Key Words:

• Flat
• Ordered monoid
• Poset
• S-poset

Mathematics Subject Classification:

• 10M50
• 06F05

ACKNOWLEDGMENTS

Bulman-Fleming research is supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494.

Gutermuth research is supported by Natural Sciences and Engineering Research Council of Canada Undergraduate Student Research Assistantship.

Gilmour research is supported by a Wilfrid Laurier University research stipend to SB-F.

Kilp research is supported by Estonian Science Foundation Grant no. 4557.

Notes

Communicated by P. Higgins.