Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532120140701Uniformities and covering properties for partial frames (I)1216481ENJohnFrithDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701,
South Africa.AnnelieseSchauerteDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.Journal Article20140410<span style="color: #000000;">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. </span>Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532120140701Uniformities and covering properties for partial frames (II)23356798ENJohnFrithDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701,
South Africa.AnnelieseSchauerteDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.Journal Article20140410This 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532120140701Quasi-projective covers of right $S$-acts37456482ENMohammadRoueentanDepartment of Mathematics, College of
Science, Shiraz University, Shiraz 71454, Iran.MajidErshadDepartment of Mathematics, College of Science, Shiraz University, Shiraz 71454, Iran.Journal Article20131227In this paper $S$ is a monoid with a left zero and $A_S$ (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. Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532120140701Dually quasi-De Morgan Stone semi-Heyting algebras I. Regularity47646483ENHanamantagouda P.SankappanavarDepartment of Mathematics, State University of New York, New Paltz, NY 12561Journal Article20140510This 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 $mathbf{RDQDStSH_1}$ 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 $mathbf{RDQDStSH_1}$-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 $mathbf{RDQDStSH_1}$. The Part II concludes with some open problems for further investigation.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532120140701Dually quasi-De Morgan Stone semi-Heyting algebras II. Regularity65826799ENHanamantagouda P.SankappanavarDepartment of Mathematics, State University of New York, New Paltz, NY 12561Journal Article20140510This 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 $mathbf{RDQDStSH_1}$ 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 $mathbf{RDQDStSH_1}$-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 $mathbf{RDQDStSH_1}$. The paper concludes with some open problems for further investigation.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532120140701Injectivity in a category: an overview on smallness conditions831126800ENM. MehdiEbrahimiDepartment of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.MahdiehHaddadiDepartment of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran.MojganMahmoudiDepartment of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.Journal Article20140601Some 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 $mod(Sigma, {mathcal E})$ of models of a set $Sigma$ of equations in a suitable category, say a Grothendieck topos ${mathcal E}$, given by M.Mehdi Ebrahimi. We close the paper by some examples to support the results.