This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.

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Power graphs: A survey

In this paper, we consider some category-theoretic properties of the
category Pos-S of all S-posets (posets equipped with a compatible right action of a pomonoid S), with monotone action-preserving maps between them. We first discuss some general category-theoretic ingredients of Pos-S; specifically, we characterize several kinds of epimorphisms and monomorphisms. Then, we present some adjoint relations of Pos-S with Pos, Set, and Act-S. In particular, we discuss free and cofree objects. We also examine other category-theoretic properties, such as cartesian closedness and monadicity. Finally, we consider projectivity in Pos-S with respect to regular epimorphisms and show that it is the same asprojectivity, although projectives are not generally retracts of free
objects over posets.

The Category of S -Posets

Preface 1. Introduction Robert Lowen and Walter Tholen 2. Monoidal structures Gavin J. Seal and Walter Tholen 3. Lax algebras Dirk Hofmann, Gavin J. Seal and Walter Tholen 4. Kleisli monoids Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal 5. Lax algebras as spaces Maria Manuel Clementino, Eva Colebunders and Walter Tholen Bibliography Tables Index.

Monoidal topology : a categorical approach to order, metric, and topology

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 ...

Indecomposable, projective, and flats-posets

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, a...

Flatness Properties of S-Posets

On Cayley graphs of rectangular groups

A topos is a category which looks and behaves very much like the category of sets, and so it may be thought of as a universe for mathematical discourses. One of the very useful topoi in many branches of mathematics as well as in computer sciences is the topos MSet, of sets with an action of a monoid M on them. It is well known that MSet, being isomorphic to the functor category Set M , is a topos. Here, we explicitly give the ingredients of a topos in MSet and investigate their properties for the working scientists and computer scientists. Among other things, we give some equivalent conditions, such as the left Ore condition, to Ω, the subobject classifier of MSet, being a Stone algera. Also the free and the cofree objects, as well as, limits and colimits are discussed in MSet.

The category of M-sets

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.

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

The notions of purity and equational compactness of universal algebras have been studied by Banaschewski and Nelson. Also, Banaschewski deals with these notions in the special case of G-sets for a group G. In this paper we study these and related concepts in the category PRO of projection algebras, that is in N
∞-sets, for the monoid N
∞ with the binary operation m.n=min {m,n}. We show that every monomorphism in PRO is pure and hence every equationally compact projection algebra is in fact injective. Then, we introduce the notions of s-purity and s-compactness by which we characterize the retractions and hence equationally compact projection algebras. And, among other results, we show that equationally compact, injective, and complete projection algebras are the same. Finally, we characterize (pure-)essential monomorphisms and construct the Equationally Compact Hulls, equivalently the Injective Hulls, of projection algebras. These results, among other things, generalize the main results of Guili, regarding completeness and s-injectivity in the category PRO

 s
 of separated projection algebras.

Mojgan Mahmoudi

Papers

The Category of S -Posets

On Cayley graphs of rectangular groups

The category of M-sets

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

Purity and Equational Compactness of Projection Algebras