Axiomatizability of the class of weakly injective <Emphasis Type="Italic">S</Emphasis>-acts

The concepts of weakly injective, fg-weakly injective, and p-weakly injective S-acts generalize that of injective S-act. We study the monoids S over which the classes of weakly injective, fg-weakly injective, and p-weakly injective S-acts are axiomatizable. We prove that the class of p-weakly injective S-acts over a regular monoid is axiomatizable.     

Regular polygons with primitive connected theories

We study the monoids S over which the class of all regular S-polygons is axiomatizable and primitive connected. We prove that the axiomatizable class of all regular S-polygons is primitive connected if and only if the semigroup R is a rectangular band of groups and R = eR for some idempotent e ∈ R, where _S R is the inclusion maximal regular subpolygon in the S-polygon _S S.     

Axiomatisability problems for S-acts, revisited

This paper discusses necessary and sufficient conditions on a monoid S, such that a class of left S-acts is first order axiomatisable. Such questions have previously been considered by Bulman-Fleming, Gould, Stepanova and others. Let $\mathcal{C}$ be a class of embeddings of right S-acts. A left S-act B is $\mathcal{C}$ -flat if tensoring with B preserves the embeddings in $\mathcal{C}$ . We find two sets (depending on a property of $\mathcal{C}$ ) of necessary and sufficient conditions on S such that the class of all $\mathcal{C}$ -flat left S-acts is axiomatisable. These results are similar to the ??replacement tossings?? results of Gould and Shaheen for S-posets. Further, we show how to axiomatise some classes using both replacement tossings and interpolation conditions, thus throwing some light on the former technique.     

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 the codimension of modules over skew power series rings with applications to Iwasawa algebras

The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second goal is the investigation of the class of S-modules which are finitely generated as R-modules. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups.     

On a conjecture of Bulman-Fleming and Gilmour

If S is a monoid, the set S×S equipped with componentwise S-action is called the diagonal act of S and is denoted by D(S). We prove the following theorem: the right S-act S ⁿ (1≠n∈?) is (principally) weakly flat if and only if \(\prod _{i=1}^{n}A_{i}\) is (principally) weakly flat where A _i , 1≤i≤n are (principally) weakly flat right S-acts, if and only if the diagonal act D(S) is (principally) weakly flat. This gives an answer to a conjecture posed by Bulman-Fleming and Gilmour (Semigroup Forum 79:298–314, 2009). Besides, we present a fair characterization of monoids S over which the diagonal act D(S) is (principally) weakly flat and finally, we impose a condition on D(S) in order to make S a left PSF monoid.     

Model-theoretic properties of regular polygons

This work is devoted to results obtained in the model theory of regular polygons. We give a characterization of monoids with axiomatizable and model-complete class of regular polygons. We describe monoids with complete class of regular polygons that satisfy some additional conditions. We study monoids whose regular core is represented as a union of finitely many principal right ideals and all regular polygons over which have a stable and superstable theory. We prove the stability of the class of all regular polygons over a monoid provided this class is axiomatizable and model-complete. We also describe monoids for which the class of all regular polygons is superstable and ω-stable provided this class is axiomatizable and model-complete. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 107–157, 2004.     

Monoids over which all strongly flat left acts are regular

Let Sbe a monoid. It is shown that all strongly flat left S-acts are regular if and only if all left S-acts having the property (E) are regular if and only if Sis a left PP monoid and satisfies (FP₂).This result answers a question in Kilp and Knauer [5].     

On pseudo-amenability of commutative semigroup algebras and their second duals

In this paper, we first characterize pseudo-amenability of semigroup algebras \(\ell ^1(S),\) for a certain class of commutative semigroups S,  the so-called archimedean semigroups. We show that for an archimedean semigroup S,  pseudo-amenability, amenability and approximate amenability of \(\ell ^1(S)\) are equivalent. Then for a commutative semigroup S,  we show that pseudo-amenability of \(\ell ^{1}(S)\) implies that S is a Clifford semigroup. Finally, we give some results on pseudo-amenability and approximate amenability of the second dual of a certain class of commutative semigroup algebras \(\ell ^1(S)\).     

Simpleness of Leavitt path algebras with coefficients in a commutative semiring

In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, establishing some fundamental properties of those algebras. We provide a complete characterization of ideal-simple Leavitt path algebras with coefficients in a commutative semiring S, extending the well-known characterizations when S is a field or a commutative ring. We also present a complete characterization of congruence-simple Leavitt path algebras over row-finite graphs with coefficients in a commutative semiring S.     

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.     

On Primitive and Realisable Classes

Let S be a scheme, and let G be a finite, flat, commutative group scheme over S. In this paper we show that (subject to a mild technical assumption) every primitive class in Pic(G) is realisable. This gives an affirmative answer to a question of Waterhouse. We also discuss applications to locally free classgroups and to Selmer groups of Abelian varieties.     

Products of idempotent endomorphisms of free acts of infinite rank

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (i.e., composite) of idempotent self-maps of that set. Using a wreath product construction introduced by V. Fleischer, the first-named author was recently able to describe products of idempotent endomorphisms of a freeS-act of finite rank whereS is any monoid. The purpose of the present paper is to extend this result to freeS-acts of infinite rank.Research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494     

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.     

Torsion and divisibility in finitely generated commutative semirings

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties—torsion and almost-divisibility (an element \(a\in S\) is called almost-divisible in S if there is \(b\in \mathbb {N}\cdot a\) such that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements \(f\in F(X)\) such that for every epimorphism \(\pi \) of F(X) torsion and almost-divisibility of \(\pi (f)\) are equivalent in \(\pi (F(X))\).     

Operads and varieties of algebras defined by polylinear identities

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.     

<Emphasis Type="Italic">e</Emphasis>-Invertible Matrices Over Commutative Semirings

In this article, the e-invertible matrices over commutative semirings are studied. Some properties and equivalent characterizations of the e-invertible matrices are given. Also, the interrelationships between invertible matrices and e-invertible matrices over commutative semirings are discussed. The main results obtained in this article generalize and enrich the corresponding results about invertible matrices over commutative semirings.