Sydney Bulman-Fleming

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.

PULLBACKS AND FLATNESS PROPERTIES OF ACTS. II

Let S be a monoid. A right S-act A is pullback flat (= strongly flat) if the functor A ⊗–(from the category of left S-acts to the category of sets) preserves pullbacks. We investigate possible generalizations of this notion, obtained either by restricting attention to certain types of pullbacks or by weakening the requirement of pullback preservation. We note that it is possible to describe the already familiar notions of flatness, (principal) weak flatness, and torsion freeness in these terms. Furthermore, a number of new properties arise.

INDECOMPOSABLE, PROJECTIVE, AND FLATS-POSETS

Abstract For a monoid S , a (left) S -act is a nonempty set B together with a mapping S ×B→B sending (s, b) to sb such that S (tb) = lpar;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. (New York: Walter de Gruyter, 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 given, which is then applied to investigate several flatness properties.

Monoids over which all flat cyclic right acts are strongly flat

ConclusionsThe table at the end of this paper summarizes the definitive results that have so far been obtained on the classification of monoids by properties of their cyclic right acts, where the properties in question belong to the collection (free, projective, strongly flat, condition (P), flat, weakly flat, torsion free). The presence of question marks in the table indicates that an exact description of the class of monoids in question has not yet been determined (although partial results may be known: for example, Theorem 3.6 of the present paper describes theleft PP monoids over which (weakly) flat cyclic right acts have property (P), [16] and [17] contain further such results.)Among the classes of monoids whose exact descriptions remain unknown if one attempts a classification by properties ofarbitrary rather than cyclic right acts are the following: {S| every rightS-act having (P) is projective} {S| every rightS-act having (P) is strongly flat} {S| every (weakly) flat rightS-act has (P)} {S| every weakly flat rightS-act is flat} {S| every rightS-act is flat}

Flatness properties of monocyclic acts

In a previous paper the authors studied flatness properties of cyclic actsS/ρ (S denotes a monoid, and ρ is a right congruence onS), and determined conditions onS under which all flat or weakly flat acts of this type are actually strongly flat or projective. In the present paper attention is restricted to monocyclic acts (cyclic acts in which ρ is generated by a single pair of elements ofS), and further results on such collapsing of flatness properties are obtained. An observation which is used extensively in this study is the fact that forw andt inS withwt≠t,S/ρ(wt,t) is flat if and only ift is a regular element ofS.

Left absolutely flat generalized inverse semigroups

A semigroup S is called (left, right) absolutely flat if all of its (left, right) S-sets are flat. S is a (left, right) generalized inverse semigroup if S is regula, and its set of idempotents E(S) is a (left, right) normal band (i.e. a strong semilattice of (left zero, right zero) rectangular bands). In this paper it is proved that a generalized inverse semigroup S is left absolutely flat if and only if S is a right generalized inverse semigroup and the (nonidentity) structure maps of E(S) are constant. In particular all inverse semigroups are left (and right) absolutely flat (see (1)). Other consequences are derived.

AXIOMATISABILITY OF WEAKLY FLAT, FLAT, AND PROJECTIVE S-ACTS

ABSTRACT This paper gives necessary and sufficient conditions on a monoid S that the class of flat left S-acts be (first order) axiomatisable. It presents also a streamlined version of the result of Stepanova that the class of projective left S-acts is axiomatisable if and only if the strongly flat left S-acts are axiomatisable and S is left perfect.

Flatness Properties of Acts over Commutative, Cancellative Monoids

This note presents a classification of commutative, cancellative monoids S by flatness properties of their associated S -acts.

Equalizers and Flatness Properties of Acts

Abstract In 1971, Stenström proved that the strongly flat right acts A S over a monoid S (that is, the acts that are directed colimits of finitely generated free acts) are those for which the functor A S ⊗ - (from the category of left S-acts into the category of sets) preserves pullbacks and equalizers. In 1991, it was shown by Bulman-Fleming that in fact pullback preservation alone was sufficient. Recently, the present authors, together with Valdis Laan, published two papers (Comm. Algebra 29(2) (2001), 829-850, 851-878) giving a spectrum of properties based on preservation of particular types of pullbacks that is sufficiently broad to capture all known notions of flatness, and to give new ones besides. The present paper initiates an analogous study for equalizer preservation properties. Again it turns out that the “standard” properties of flatness, weak flatness, principal weak flatness, and torsion freeness can all be described in terms of equalizer preservation, and that new properties arise as well.

On V. Fleischer's characterization of absolutely flat monoids

In his paperCompletely flat monoids (Učh. Zap. Tartu Un-ta610 (1982), 38–52 (Russian)) V. Fleischer gives a characterization of the absolute flatness of a monoidS in terms of certain one-sided ideals and one-sided congruences ofS. In the present work an alternative, more direct proof of Fieischers theorem is provided, and the result is used to show that the multiplicative monoid of any semisimple Artinian ring is absolutely flat.