The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

Abstract Given a monoid S acting (on the left) on a set X, all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X. Conversely, we prove that any Alexandroff topology may be obtained through a monoid action. Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions. Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets. Then, in relation to such a dependence, we study the family 𝒜σ,X{\\mathcal{A}_{\\sigma,X}} of the closed subsets Y of X such that, for any y1,y2∈Y{y_{1},y_{2}\\in Y}, there exists a third element y∈Y{y\\in Y} whose closure contains both y1{y_{1}} and y2{y_{2}}. More in detail, relying on some specific properties of the maximal members of the family 𝒜σ,X{\\mathcal{A}_{\\sigma,X}}, we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence. Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions.

/pdf/alexandroff-topologies-and-monoid-actions-4yfcx9zoxf.pdf

Alexandroff topologies and monoid actions

In 1981 Edgar Enochs conjectured that every module has a flat cover and finally proved this in 2001. Since then a great deal of effort has been spent on studying different types of covers, for example injective and torsion free covers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids but their definition of cover was slightly different from that of Enochs. Recently, Bailey and Renshaw produced some preliminary results on the `other' type of cover and it is this work that is extended in this paper. We consider free, divisible, torsion free and injective covers and demonstrate that in some cases the results are quite different from the module case.

Covers of acts over monoids II

In 1981 Edgar Enochs conjectured that every module over a unitary ring has a flat cover. He finally proved this conjecture in 2001, in a paper that included an independent proof by Bican and El Bashir. Enochs had in fact considered different types of covers as early as 1963, for example injective and torsion free covers, and since then a great deal of effort has been spent on their study. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids but their definition of cover was slightly different from that of Enochs. Recently, Bailey and Renshaw produced some preliminary results on the ‘other’ type of cover and it is this work that is extended in this paper. We consider free, divisible, torsion free and injective covers and demonstrate that in some cases the results are quite different from the module case.

/pdf/covers-of-acts-over-monoids-ii-3hun3o6dbn.pdf

The concept of a weak factorization system has been studied extensively in homotopy theory and has recently found an application in one of the proofs of the celebrated flat cover conjecture, categorical versions of which have been presented by a number of authors including Rosický. One of the main aims of this paper is to draw attention to this interesting concept and to initiate a study of these systems in relation to flatness of S-acts and related concepts.

/pdf/weak-factorization-systems-for-s-acts-1q769rdnbw.pdf

Weak factorization systems for S -acts

In 2001 Enochs' celebrated flat cover conjecture was finally proven and the proofs (two different proofs were presented in the same paper [5]) have since generated a great deal of interest among researchers. In particular the results have been recast in a number of other categories and in particular for additive categories (see for example [2], [3], [23] and [24]). In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that in general their definition was not equivalent to Enochs', except in the projective case, and left open a number of questions regarding the `other' definition. This `other' definition is the subject of the present paper and we attempt to emulate some of Enochs' work for the category of acts over monoids and concentrate, in the main, on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion free, divisible and free, in a future report.

/pdf/covers-of-acts-over-monoids-and-pure-epimorphisms-3v22ha5t6s.pdf

Covers of Acts Over Monoids and Pure Epimorphisms

In 2001 Enoch's celebrated flat cover conjecture was finally proven and the proofs (two different proofs were presented in the same paper [4]) have since generated a great deal of interest among researchers. In particular the results have been recast in a number of other categories and in particular for additive categories (see for example [2], [3], [22] and [23]). In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that in general their definition was not equivalent to Enoch's, except in the projective case, and left open a number of questions regarding the `other' definition. This `other' definition is the subject of the present paper and we attempt to emulate some of Enoch's work to the category of acts over monoids and concentrate in the main on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion free, divisible and free, in a future report.

Covers of acts over monoids and pure epimorphisms

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Covers of Acts Over Monoids and Pure Epimorphisms

Covers of acts over monoids and pure epimorphisms

Covers of acts over monoids II

Covers of acts over monoids II

Weak factorization systems for S -acts