On the homological classification of pomonoids by their Rees factor S-posets

The study of flatness properties of pomonoids acting on posets was initiated by S.M. Fakhruddin in the 1980s. This work has recently been continued by various authors (see references). The Rees factor S-posets are investigated in S. Bulman-Fleming, D. Gutermuth, A. Gilmour and M. Kilp, Flatness properties of  S -posets (Commun. Algebra 34:1291–1317, 2006). In the present article, we investigate the homological classification problems of pomonoids by their Rees factor S-posets. Supported by Research Supervisor Program of Education Department of Gansu Province (0801-03) and nwnu-kjcxgc-03-51.     

The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.     

On monoids over which all generators satisfy a flatness property

This paper addresses conditions under which all generators in the category of right S-acts (where S is a monoid) satisfy a flatness property. There are characterizations for monoids over which all generators satisfy a flatness property α where α can stand for freeness, projectivity, strong flatness, Condition (P), principal weak flatness and torsion freeness. To our knowledge, the problem has not been studied for other flatness properties such as weak flatness, Condition (E) and regularity. The present paper addresses this gap.     

On strongly flat and condition (P) <Emphasis Type="Italic">S</Emphasis>-posets

In this paper, we first present some homological classifications of pomonoids by using condition (P) and strongly flat properties. Unlike the case for acts, condition (P) and strongly flat coincide for cyclic right S-posets when all weakly right reversible convex subpomonoids of a pomonoid S are left collapsible. Thereby we characterize pomonoids over which strong flatness and condition (P) imply some other flatness properties. Furthermore, we characterize a pomonoid over which every right S-poset has a strongly flat (condition (P)) cover.     

On the homological classification of pomonoids by properties of cyclic S-posets

Between different and relatively well investigated so-called flatness properties of S-posets there is a property called property ( ${\rm P}_{w}$ ) which, so far, has not received much attention. In this paper, we characterize pomonoids from a subclass of completely simple semigroups with adjoined identity all of whose cyclic (Rees factor) S-posets satisfy property ( ${\rm P}_{w}$ ). Moreover, for the same class of pomonoids, we find necessary and sufficient conditions under which all Rees factor S-posets satisfying property ( ${\rm P}_{w}$ ) satisfy property (P).     

Absolute flatness and amalgamation in pomonoids

Absolute flatness and amalgamation for partially ordered monoids (briefly pomonoids) were first considered in the mid 1980s by S.M. Fakhruddin in two research articles. Though the study of absolute flatness for pomonoids was revived by X. Shi, S. Bulman-Fleming and others after a dormancy period of almost two decades—resulting in the appearance of several research articles on the subject since 2005—amalgamation in pomonoids was never reconsidered until the recent past when S. Bulman-Fleming and the author produced two research articles on the subject. The primary objectives of these papers were to show that imposition of order subjects the amalgamation of monoids to severe restrictions and to prove that partially ordered groups (briefly pogroups) are amalgamation bases in the class of all pomonoids. Proceeding further, we establish in this article the amalgamation property for the class of pogroups. (The property was first proved for the class of groups by O. Schreier, Abh. Math. Semin. Univ. Hamb. 5:161–183, 1927.) In addition, we show that absolutely flat commutative pomonoids are (strong) amalgamation bases in the category of commutative pomonoids. (A similar result was proved by Fakhruddin for weak amalgamation.) The special amalgamation property and the existence of pushouts in the category of pomonoids, which have been instrumental in proving our main results, are also established.     

Principally Weakly and Weakly Coherent Monoids

We shall call a monoid S principally weakly (weakly) left coherent if direct products of nonempty families of principally weakly (weakly) flat right S-acts are principally weakly (weakly) flat. Such monoids have not been studied in general. However, Bulman-Fleming and McDowell proved that a commutative monoid S is (weakly) coherent if and only if the act S ^I is weakly flat for each nonempty set I. In this article we introduce the notion of finite (principal) weak flatness for characterizing (principally) weakly left coherent monoids. Also we investigate monoids over which direct products of acts transfer an arbitrary flatness property to their components.     

Equivariant completeness and regular injectivity of S-posets

Abstract

In this paper, considering the actions of a pomonoid S on posets, namely S-posets, we study some relations between equivariant completeness and regular injectivity of S-posets which lead to some homological classification results for pomonoids. In particular, we show that regular injectivity implies equivariant completeness, but the converse is true only if S is left simple. Finally, it is proved that regularly injective S-posets are exactly the complete and cofree-retract ones. Among other results, we also see that the Skornjakov and Baer criteria fail for regular injectivity of S-posets.     

Equalizers and Flatness Properties of Acts,II

In Comm. Algebra 30 (3) (2002), 1475–1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act A_S over a monoid S that are based on the extent to which the functor A_S$\otimes -$ preserves equalizers. In Semigroup Forum 65 (3) (2002), 428–449, Bulman-Fleming discussed in detail one of these notions, annihilator-flatness. The present paper is devoted to two more of these notions, weak equalizer-flatness and strong torsion-freeness. An act A_S is called weakly equalizer-flat if the functor A_S$\otimes -$ almost preserves equalizers of any two homomorphisms into the left act _SS, and strongly torsion-free if this functor almost preserves equalizers of any two homomorphisms from _SS into the Rees factor act _S(S/Sc), where c is any right cancellable element of S. (The adverb almost signifies that the canonical morphism provided by the universal property of equalizers may be only a monomorphism rather than an isomorphism.) From the definitions it is clear that flatness implies weak equalizer-flatness, which in turn implies annihilator-flatness, and it was known already that both of these implications are strict. A monoid is called right absolutely weakly equalizer-flat if all of its right acts are weakly equalizer-flat. In this paper we prove a result which implies that right PP monoids with central idempotents are absolutely weakly equalizer-flat. We also show that for a relatively large class of commutative monoids, right absolute equalizer-flatness and right absolute annihilator-flatness coincide. Finally, we provide examples showing that the implication between strong torsion-freeness and torsion-freeness is strict.     

The Structure of the Reduced Product of Preinjective Modules Over a Hereditary Right Pure Semisimple Ring

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.     

Axiomatizability of free <Emphasis Type="Italic">S</Emphasis>-posets

In this work, we investigate the partially ordered monoids S over which the class of free (over a poset) S-posets is axiomatizable. Similar questions for S-sets were considered in papers of V. Gould, S. Bulman-Fleming, and A. A. Stepanova.     

Nilpotent Elements and Idempotents in Commutative Group Rings #

Flatness properties of acts over monoids and their connection with monoid amalgamation have been investigated for almost four decades and a substantial literature on the subject has now appeared. Analogous research, concerning the action of partially ordered monoids on partially ordered sets and its relation to pomonoid amalgamation, was begun in 1980s in two articles by S. M. Fakhruddin. The subject then remained dormant until the recent past when several articles on flatness in the setting of ordered monoids acting on posets (briefly, S-posets) appeared. It has now been established that the introduction of order results in severe restrictions as far as absolute flatness is concerned. Also, after formulating in the ordered context the Representation Extension and Right Congruence Extension Properties, first used by T. E. Hall to study semigroup amalgams, the authors recently observed in their article [4 Bulman-Fleming , S. , Nasir , S. ( 2010 ). Examples concerning absolute flatness and amalgamation in pomonoids . Semigroup Forum 80 : 272 – 292 .[Crossref], [Web of Science ®] , [Google Scholar]] that inverse monoids, though being amalgamation bases in the class of all monoids, may fail to possess this property when put in the ordered scenario. The purpose of the present article is to formulate an ordered version of the Strong Representation Extension Property of Hall and to explore its connections with absolute flatness and amalgamation of pomonoids. This enables us to prove that pogroups are (strong) amalgamation bases in the class of all pomonoids.     

Banaschewski’s theorem for S-posets: regular injectivity and completeness

In this paper we study the notion of injectivity in the category Pos-S of S-posets for a pomonoid S. First we see that, although there is no non-trivial injective S-poset with respect to monomorphisms, Pos-S has enough (regular) injectives with respect to regular monomorphisms (sub S-posets). Then, recalling Banaschewski’s theorem which states that regular injectivity of posets with respect to order-embeddings and completeness are equivalent, we study regular injectivity for S-posets and get some homological classification of pomonoids and pogroups. Among other things, we also see that regular injective S-posets are exactly the retracts of cofree S-posets over complete posets.     

Axiomatisability problems for <Emphasis Type="Italic">S</Emphasis>-posets

Let \(\mathcal{C}\) be a class of ordered algebras of a given fixed type τ. Associated with the type is a first order language L _τ , which must also contain a binary predicate to be interpreted by the ordering in members of \(\mathcal{C}\). One can then ask the question, when is the class \(\mathcal{C}\) axiomatisable by sentences of L _τ ? In this paper we will be considering axiomatisability problems for classes of left S-posets over a pomonoid S (that is, a monoid S equipped with a partial order compatible with the binary operation). We aim to determine the pomonoids S such that certain categorically defined classes are axiomatisable. The classes we consider are the free S-posets, the projective S-posets and classes arising from flatness properties. Some of these cases have been studied in a recent article by Pervukhin and Stepanova. We present some general strategies to determine axiomatisability, from which their results for the classes of weakly po-flat and po-flat S-posets will follow. We also consider a number of classes not previously examined.     

Completion of S-posets

In this paper, analogously to the construction of completion of pomonoids, we construct a completion of S-posets for a pomonoid?S, which is compatible with joins (supremums).     

Subpullback flat S-posets need not be subequalizer flat

If S is a monoid, a right S-act A _S is a set A, equipped with a “right S-action” A×S→A sending the pair (a,s)∈ A×S to as, that satisfies the conditions (i) a(st)=(as)t and (ii) a1=a for all a∈A and s,t∈S. If, in addition, S is equipped with a compatible partial order and A is a poset, such that the action is monotone (when A×S is equipped with the product order), then A _S is called a right S-poset. Left S-acts and S-posets are defined analogously. For a given S-act (resp. S-poset) a tensor product functor A _S ?? from left S-acts to sets (resp. left S-posets to posets) exists, and A _S is called pullback flat or equalizer flat (resp. subpullback flat or subequalizer flat) if this functor preserves pullbacks or equalizers (resp. subpullbacks or subequalizers). By analogy with the Lazard-Govorov Theorem for R-modules, B. Stenström proved in 1971 that an S-act is isomorphic to a directed colimit of finitely generated free S -acts if and only if it is both pullback flat and equalizer flat. Some 20 years later, the present author showed that, in fact, pullback flatness by itself is sufficient. (A new, more direct proof of that result is contained in the present article.) In 2005, Valdis Laan and the present author obtained a version of the Lazard-Govorov Theorem for S-posets, in which subpullbacks and subequalizers now assume the role previously played by pullbacks and equalizers. The question of whether subpullback flatness implies subequalizer flatness remained unsolved. The present paper provides a negative answer to this question.     

On Characterization of Monoids by Properties of Generators

Kilp and Knauer in(Comm. Algebra, 1992, 20(7), 1841–1856) gave characterizations of monoids when all generators in category of right S-acts(S is a monoid) satisfy properties such as freeness, projectivity, strong flatness, Condition(P), principal weak flatness, principal weak injectivity, weak injectivity, injectivity, divisibility, strong faithfulness and torsion freeness.Sedaghtjoo in(Semigroup Forum, 2013, 87: 653–662) characterized monoids by some other properties of generators including weak flatness, Condition(E) and regularity. To our knowledge,the problem has not been studied for properties mentioned above of(finitely generated, cyclic,monocyclic, Rees factor) right acts. In this article we answer the question corresponding to these properties and also f g-weak injectivity.     

On Morita Equivalence of Partially Ordered Monoids

We show that there is one-to-one correspondence between certain algebraically and categorically defined subobjects, congruences and admissible preorders of S-posets. Using preservation properties of Pos-equivalence functors between Pos-categories we deduce that if S and T are Morita equivalent partially ordered monoids and F:Pos _S →Pos _T is a Pos-equivalence functor then an S-poset A _S and the T-poset F(A _S ) have isomorphic lattices of (regular, downwards closed) subobjects, congruences and admissible preorders. We also prove that if A _S has some flatness property then F(A _S ) has the same property.     

Some 3/2-transitive permutation groups in which the stabilizer of two points is always Abelian

A monoidS is susceptible to having properties bearing upon all right acts overS such as: torsion freeness, flatness, projectiveness, freeness. The purpose of this note is to find necessary and sufficient conditions on a monoidS in order that, for example, all flat rightS-acts are free. We do this for all meaningful variants of such conditions and are able, in conjunction with the results of Skornjakov [8], Kilp [5] and Fountain [3], to describe the corresponding monoids, except in the case all torsion free acts are flat, where we have only some necessary condition. We mention in passing that homological classification of monoids has been discussed by several authors [3, 4, 5, 8].In the following,S will always stand for a monoid. A rightS-act is a setA on whichS acts unitarily from the right in the usual way, that is to saya(rs) = (ar)s, a1 =a (a A,r,s S) where 1 denotes the identity ofS.