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.     

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 flatness properties of cyclic S-posets

In this paper, we discuss flatness properties of cyclic S-posets. As applications, some partially ordered monoids are characterized and some results on S-acts can be also obtained. Research supported by the National Natural Science Foundation of China (No.10626012).     

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.     

Regular S-acts with primitive normal and antiadditive theories

In this work, we investigate the commutative monoids over which the axiomatizable class of regular S-acts is primitive normal and antiadditive. We prove that the primitive normality of an axiomatizable class of regular S-acts over the commutative monoid S is equivalent to the antiadditivity of this class and it is equivalent to the linearity of the order of a semigroup R such that an S-act sR is a maximal (under the inclusion) regular subact of the S-act sS.     

Monoids for Which Condition (P) Acts are Projective

A characterisation of monoids for which all right S-acts satisfying conditions (P) are projective is given. We also give a new characterisation of those monoids for which all cyclic right S-acts satisfying condition (P) are projective, similar in nature to recent work by Kilp [6]. In addition we give a sufficient condition for all right S-acts that satisfy condition (P) to be strongly flat and show that the indecomposable acts that satisfy condition (P) are the locally cyclic acts.     

On Monoids for Which all Condition (P) Acts are Strongly Flat

In this article, we continue to investigate the monoids over which all right S-acts satisfying condition (P) are strongly flat, and we obtain some new classes of monoids, thereby extending all previous results in this area.     

Perfection for pomonoids

A pomonoid S is a monoid equipped with a partial order that is compatible with the binary operation. In the same way that M-acts over a monoid M correspond to the representation of M by transformations of sets, S-posets correspond to the representation of a pomonoid S by order preserving transformations of posets.     

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.     

On Some Finitary Conditions Arising from the Axiomatisability of Certain Classes of Monoid Acts

This article considers those monoids S satisfying one or both of the finitary properties (R) and (r), focussing for the most part on inverse monoids. These properties arise from questions of axiomatisability of classes of S-acts, and appear to be of interest in their own right. If S is weakly right noetherian (WRN), that is, S has the ascending chain condition on right ideals, then certainly (r) holds. Other than this, we show that (R), (r), and (WRN) are independent. Our most detailed results are for Clifford monoids, in which case we completely characterise those S with trivial structure homomorphisms satisfying (R) or (r).     

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).     

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.     

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.     

Quasi‐completeness and functions without fixed‐points

We prove a completeness criterion for quasi‐reducibility and generalize it to higher levels of the arithmetical hierarchy. As an application of the criterion we obtain Q‐completeness of the set of all pairs (x, n) such that the prefix‐free Kolmogorov complexity of x is less than n. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems

In this paper we obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by Hayden and Winter to get violations of the additivity equalities for minimum output Rényi entropies. We show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the p-Rényi entropy, for all p>1. For some sequences of random quantum channels, we compute almost surely the limit of their Schatten S₁→Sp norms.     

A NOTION OF COMPLETENESS OF TOPOLOGICAL STRUCTURES

A connector U on a space S is a function from S to the power set of S such that each x in s belongs to its image. The image of x is denoted by xU. In other words, the relation {(x,y): y ? xU, x ? S) is a reflexive binary relation. A space with a certain set of connectors is a generalization of topological spaces as well as uniform spaces. In this paper, a notion of completeness of such a space is introduced. This completeness corresponds to completeness of uniform spaces if a set of cannectors meets the conditions of uniformity. Compactness of topological Spaces is a special case of the completeness.     

Free <Emphasis Type="Bold">C</Emphasis><Subscript>+</Subscript>-actions on <Emphasis Type="Bold">C</Emphasis><Superscript>3</Superscript> are translations

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C₊-action on it such that S=X//C₊ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C₊-action on C³ is a translation in a suitable coordinate system.     

Projective S-acts and Exact Functors

Let S be a semigroup. In this paper, projective S-acts and exact sequences in S-Act are studied. It is shown that, for a unitary S-act P, the functor Hom(P, –) is exact if and only if P Se for some idempotent e S.The research was supported by the National Natural Science Foundation of China.1991 Mathematics Subject Classification: 20M50     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

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.