Categories and General Algebraic Structures with Applications
Volume:2 Issue: 1, Jul 2014

Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive lattices, sigma-frames, kappa-frames and frames. Reflective subcategories of uniform and nearness spaces and lately coreflective subcategories of uniform and nearness frames have been a topic of considerable interest. In cite{jfas9} an easily implementable criterion for establishing certain coreflections in nearness frames was presented. Although the primary application in that paper was in the setting of nearness frames, it was observed there that similar techniques apply in many categories; we establish here, in this more general setting of structured partial frames, a technique that unifies these. We make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting our axiomatization of partial frames, which we call sels-frames, we add structure, in the form of sels-covers and nearness, and provide the promised method of constructing certain coreflections. We illustrate the method with the examples of uniform, strong and totally bounded nearness sels -frames. In Part (II) of this paper, we consider regularity, normality and compactness for partial frames.

This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which we make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting there our axiomatization of partial frames, which we call sels-frames, we added structure, in the form of sels-covers and nearness. Here, in the unstructured setting, we consider regularity, normality and compactness, expressing all these properties in terms of sels-covers. We see that an sels-frame is normal and regular if and only if the collection of all finite sels-covers forms a basis for an sels-uniformity on it. Various results about strong inclusions culminate in the proposition that every compact, regular sels-frame has a unique compatible sels -uniformity.

In this paper S is a monoid with a left zero and AS (or A) is a unitary right S-act. It is shown that a monoid S is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right S-act is quasi-projective. Also it is shown that if every right S -act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that every right act has a projective cover. ý

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended lor-De Morgan law introduced in cite{Sa12}. Then, using this result and the results of cite{Sa12}, we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) in the variety of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. It is shown that there are 25 nontrivial simple algebras in this variety. In Part II, we prove, using the description of simples obtained in this Part, that the variety mathbfRDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element mathbfRDQDStSH1-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras--the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of this theorem, we present (equational) axiomatizations for several subvarieties of mathbfRDQDStSH1 . The Part II concludes with some open problems for further investigation.

This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety mathbfRDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element mathbfRDQDStSH1-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras--the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of our main theorem, we present (equational) axiomatizations for several subvarieties of mathbfRDQDStSH1 . The paper concludes with some open problems for further investigation.

Some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness. In this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by Bernhard Banaschewski and Walter Tholen, who studied these notions in a much more categorical setting. Then, we study these notions as well as the well behavior of injectivity, in the class mod(Sigma,mathcalE) of models of a set Sigma of equations in a suitable category, say a Grothendieck topos mathcalE , given by M.Mehdi Ebrahimi. We close the paper by some examples to support the results.