Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240101Celebrating Professor Themba A. Dube (A TAD Celebration I)3310310422910.48308/cgasa.2023.234071.1453ENInderasan NaidooDepartment of Mathematical Sciences, University of South Africa, P.O. Box 392, Tshwane, UNISA 0003, South Africa.\\
National Institute for Theoretical and Computational Sciences (NITheCS), Johannesburg, South
Africa.0000-0002-3454-2268Journal Article20231208This is the first in a series of survey papers featuring the mathematical contributions of Themba Dube to pointfree topology and ordered algebraic structures. We cover Dube’s distinguished career and benefactions to the discipline with the early beginnings in nearness frames. We envelope the essential aspects of Dube’s work in structured frames. The paper radars across the initial themes of nearness, metrization, and uniform structures that Dube conceives and presents in his independent and joint published papers. Pertinent subcategories of these structured frames are discussed. We also feature Dube’s imprints on certain categorical aspects of his work on βL, λL, υL and ßL.https://cgasa.sbu.ac.ir/article_104229_b8136648876f7971825ceeaba208e6e8.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240101Notes on the spatial part of a frame10512910413810.48308/cgasa.2023.233584.1435ENIgor ArrietaSchool of Computer Science, University of Birmingham, B15 2TT
Birmingham, UK0000-0002-5319-4916Jorge PicadoDepartment of Mathematics
University of Coimbra
PORTUGAL0000-0001-7837-1221Ales PultrDepartment of Applied Mathematics and ITI, MFF, Charles University,
Malostranské ném. 24, 11800 Praha 1, Czech Republic0000-0002-9308-3700Journal Article20231027A locale (frame) L has a largest spatial sublocale generated by the primes (spectrum points), the spatial part SpL. In this paper we discuss some of the properties of the embeddings SpL ⊆ L. First we analyze the behaviour of the spatial parts in the assembly: the points of L and of S(L)^op (∼=<br />the congruence frame) are in a natural one-one correspondence while the topologies of SpL and Sp(S(L)^op) differ. Then we concentrate on some special types of embeddings of SpL into L, namely in the questions when SpL is complemented, closed, or open. While in the first part L was general, here we need some restrictions (weak separation axioms) to obtain suitable formulashttps://cgasa.sbu.ac.ir/article_104138_20328dd2785ec1b9344ad14f61b4c907.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240101$\alpha$-Projectable and laterally $\alpha$-complete Archimedean lattice-ordered groups with weak unit via topology13115410408710.48308/cgasa.2023.234039.1448ENBrian WynneDepartment of Mathematics, Lehman College, City University of New York, Bronx, USA0000-0002-4043-2508Anthony WoodHagerDepartment of Mathematics and CS, Wesleyan University, Middletown, CT 06459.Journal Article20231206Let $\bf{W}$ be the category of Archimedean lattice-ordered groups with weak order unit, $\bf{Comp}$ the category of compact Hausdorff spaces, and $\mathbf{W} \xrightarrow{Y} \mathbf{Comp}$ the Yosida functor, which represents a $\bf{W}$-object $A$ as consisting of extended real-valued functions $A \leq D(YA)$ and uniquely for various features. This yields topological mirrors for various algebraic ($\bf{W}$-theoretic) properties providing close analysis of the latter. We apply this to the subclasses of $\alpha$-projectable, and laterally $\alpha$-complete objects, denoted $P(\alpha)$ and $L(\alpha)$, where $\alpha$ is a regular infinite cardinal or $\infty$. Each $\bf{W}$-object $A$ has unique minimum essential extensions $A \leq p(\alpha) A \leq l(\alpha) A$ in the classes $P(\alpha)$ and $L(\alpha)$, respectively, and the spaces $Yp(\alpha) A$ and $Yl(\alpha) A$ are recognizable (for the most part); then we write down what $p(\alpha) A$ and $l(\alpha) A$ are as functions on these spaces. The operators $p(\alpha)$ and $l(\alpha)$ are compared: we show that both preserve closure under all implicit functorial operations which are finitary. The cases of $A = C(X)$ receive special attention. In particular, if ($\omega < \alpha$) $l(\alpha) C(X) = C(Yl(\alpha) C(X))$, then $X$ is finite. But ($\omega \leq \alpha$) for infinite $X$, $p(\alpha) C(X)$ sometimes is, and sometimes is not, $C(Yp(\alpha) C(X))$.https://cgasa.sbu.ac.ir/article_104087_3473b026483ef5fc2580f37eb7c9f9a2.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240101S-Metrizability and the Wallman basis of a frame15517410409410.48308/cgasa.2023.233801.1440ENCerene RathilalUniveristy of KwaZulu-Natal0000-0002-9026-7547Journal Article20231117The Wallman basis of a frame and the corresponding induced compactification was first investigated by Baboolal [2]. In this paper, we provide an intrinsic characterisation of S-metrizability in terms of the Wallman basis of a frame. Particularly, we show that a connected, locally connected frame is S-metrizable if and only if it has a countable locally connected and uniformly connected Wallman basis.<br /><br />https://cgasa.sbu.ac.ir/article_104094_c3ec945a4b27032025005b28513c58cc.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240101A little more on ideals associated with sublocales17520010410210.48308/cgasa.2023.234093.1456ENOghenetega IghedoDepartment of Mathematics, Chapman University, P.O. Box 92866, California, U.S.A.0000-0001-7968-6006Grace Wakesho KivungaDepartment of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa.Dorca Nyamusi StephenDeparment of Mathematics and Physics, Technical University of Mombasa, P.O. Box 90420-80100, Mombasa, Kenya.Journal Article20231211 As usual, let $\mathcal RL$ denote the ring of real-valued continuous functions on a completely regular frame $L$. Let $\beta L$ and $\lambda L$ denote the Stone-\v{C}ech compactification of $L$ and the Lindel\"of coreflection of $L$, respectively. There is a natural way of associating with each sublocale of $\beta L$ two ideals of $\mathcal RL$, motivated by a similar situation in $C(X)$. In~\cite{DS1}, the authors go one step further and associate with each sublocale of $\lambda L$ an ideal of $\mathcal RL$ in a manner similar to one of the ways one does it for sublocales of $\beta L$. The intent in this paper is to augment~\cite{DS1} by considering two other coreflections; namely, the realcompact and the paracompact coreflections.\\<br /> We show that $\boldsymbol M$-ideals of $\mathcal RL$ indexed by sublocales of $\beta L$ are precisely the intersections of maximal ideals of $\mathcal RL$. An $\boldsymbol{M}$-ideal of $\mathcal RL$ is \emph{grounded} in case it is of the form $\boldsymbol{M}_S$ for some sublocale $S$ of $L$. A similar definition is given for an $\boldsymbol{O}$-ideal of $\mathcal RL$. We characterise $\boldsymbol M$-ideals of $\mathcal RL$ indexed by spatial sublocales of $\beta L$, and $\boldsymbol O$-ideals of $\mathcal RL$ indexed by closed sublocales of $\beta L$ in terms of grounded maximal ideals of $\mathcal RL$.https://cgasa.sbu.ac.ir/article_104102_2ad4eee6ea89fcd4ab8026764c4c6e92.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240102On one-local retract in modular metrics20122010414610.48308/cgasa.2023.234064.1451ENOliver Olela OtafuduSchool of Mathematical and Statistical Sciences
North-West University, Potchefstroom Campus,
Potchefstroom 2520,
South Africa.0000-0001-9593-7899Tlotlo Odacious PhaweSchool of Mathematical and Statistical Sciences
North-West University, Potchefstroom Campus,
Potchefstroom 2520,
South Africa.0000-0003-2837-8147Journal Article20231208We continue the study of the concept of one local retract in the settings of modular metrics. This concept has been studied in metric spaces and quasi-metric spaces by different authors with different motivations. In this article, we extend the well-known results on one-local retract in metric point of view to the framework of modular metrics. In particular, we show that any self-map $\psi: X_w \longrightarrow X_w$ satisfying the property $w(\lambda,\psi(x),\psi(y)) \leq w(\lambda,x,y)$ for all $x,y \in X$ and $\lambda >0$, has at least one fixed point whenever the collection of all $q_w$-admissible subsets of $X_{w}$ is both compact and normal.https://cgasa.sbu.ac.ir/article_104146_511c297d30e90fabece4a56eab1d2ef8.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240101Direct products of cyclic semigroups and left zero semigroups in $\beta\mathbb{N}$22123210416210.48308/cgasa.2023.233625.1436ENYuliya ZelenyukSchool of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa.0000-0003-4741-8327Journal Article20231030We show that for every $n\in\mathbb{N}$, the direct product of the cyclic semigroup of order $n$ and period $1$ and the left zero semigroup $2^\mathfrak{c}$ has copies in $\beta\mathbb{N}$.https://cgasa.sbu.ac.ir/article_104162_494206869b9b3eca4b1c1491a735fcac.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585320120240101Topological spaces versus frames in the topos of $M$-sets23326010410510.48308/cgasa.2023.234111.1455ENMojgan MahmoudiMojgan Mahmoudi;
Department of Mathematics, Faculty of Mathematical Sciences,
Shahid Beheshti University, Tehran 19839, Iran0000-0002-7556-8536Amir H. NejahDepartment of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, IranJournal Article20231211In this paper we study topological spaces, frames, and their confrontation in the presheaf topos of $M$-sets for a monoid $M$. We introduce the internalization, of the frame of open subsets for topologies, and <br />of topologies of points for frames, in our universe. <br />Then we find functors between the categories of topological spaces and of frames in our universe.<br />We show that, in contrast to the classical case, the obtained functors do not have an adjoint relation for a general monoid, but in some cases such as when $M$ is a group, they form an adjunction. <br />Furthermore, we define and study soberity and spatialness for our topological spaces and frames, respectively. It is shown that if $M$ is a group then the restriction of the adjunction to sober spaces and spatial frames becomes into an isomorphism.https://cgasa.sbu.ac.ir/article_104105_8ef864d498af1086a8d29125553460a3.pdf