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.     

Essentiality and injectivity relative to sequential purity of acts

For a class ℳ of monomorphisms of a category, mathematicians consider different types of essentiality, depending on ℳ. In this paper, considering the category of acts over a semigroup, we first briefly study the class ℳ _p of a certain kind of pure monomorphisms, in a manner borrowed from V. Gould, to be called sequentially pure. Then, we study in detail three kinds of essentiality with respect to this class, and give some useful criteria to get (internal) characterizations (in terms of elements) for essentialities. Finally, the relations between injectivity, essentiality, retractness, and injective hulls, all with respect to the class of sequentially pure monomorphisms, are investigated. The second author is thankful to Iran National Science Foundation (INSF) for their financial support.     

Tensor products and preservation of limits, for acts over monoids

In 1971, Stenström published one of the first papers devoted to the problem of when, for a monoid S and a right S -act A _S , the functor A? (from the category of left acts over S into the category of sets) has certain limit preservation properties. Attention at first focused on when this functor preserves pullbacks and equalizers but, since that time, a large number of related articles have appeared, most having to do with when this functor preserves monomorphisms of various kinds. All of these properties are often referred to as flatness properties of acts . Surprisingly, little attention has so far been paid to the obvious questions of when A _S ? preserves all limits, all finite limits, all products, or all finite products. The present article addresses these matters.     

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.     

Characterizing Semigroups by Sequentially Dense Injective Acts

Sequentially dense monomorphisms and injectivity with respect to these monomorphisms were first introduced and studied by Giuli for acts over the monoid (N^∞, min). In this paper we generalize these notions to acts over a general semigroup, and study the behaviour of this notion of injectivity with respect to products, coproducts, and direct sums. As a result we give some characterizations of semigroups.     

Flatness properties of diagonal acts over monoids

If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S ^I (for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness. Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats, etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems. S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494. Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007).     

Rugged modules: The opposite of flatness

Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M⁺ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.     

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.     

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.     

Principal Weak Flatness and Regularity of Diagonal Acts

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ? left P(P) ? weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 _X , D(S) is regular.     

On unitary down-closed regular monomorphisms of pomonoid actions

Abstract

In this paper we characterize injective objects in the category of S-posets and S-poset maps for a pomonoid S, with respect to the class of unitary down-closed embeddings. Also, the behaviour of this notion of injectivity with respect to products and coproducts is studied. Then we introduce the notion of weakly regular d-injectivity in arbitrary slices of the category of S-posets, which is applied to investigate the Baer criterion. Finally we present an example to show that these objects are not regular injective, in general.     

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.     

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.     

Flatness properties of monocyclic acts

In a previous paper the authors studied flatness properties of cyclic actsS/ (S denotes a monoid, and 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 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 withwtt,S/(wt,t) is flat if and only ift is a regular element ofS.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A4494.Research supported by Estonian Research Foundation Grant No. 930.     

On strongly (<Emphasis Type="Italic">P</Emphasis>)-cyclic acts

By a regular act we mean an act such that all its cyclic subacts are projective. In this paper we introduce strong (P)-cyclic property of acts over monoids which is an extension of regularity and give a classification of monoids by this property of their right (Rees factor) acts.