Categories and General Algebraic Structures with Applications
https://cgasa.sbu.ac.ir/
Categories and General Algebraic Structures with Applicationsendaily1Mon, 01 Jan 2024 00:00:00 +0330Mon, 01 Jan 2024 00:00:00 +0330Cover for Vol. 20, No. 1.
https://cgasa.sbu.ac.ir/article_104226.html
Preface for Vol. 20, No. 1.
https://cgasa.sbu.ac.ir/article_104227.html
An interview with Themba Andrew Dube (A TAD Interview)
https://cgasa.sbu.ac.ir/article_104172.html
The paper covers an interview with Professor Themba Andrew Dube that captures his academic career that spans ordered algebraic structures, frames (locales), category theory, and pointfree function rings. The dialogue between the author and Dube features some of the work of Dube's doctoral students that influenced his research direction. The collaborative work of Dube and his relationships with prominent local and international scholars is exchanged in the conversation. We also narrate Dube's academic citizenry in Quaestiones Mathematicae, the journal of the South African Mathematical Society, whilst he was Editor-in-Chief. The paper gives a valuable historical account of Dube's contribution to the South African mathematical landscape and the African mathematical diaspora.Celebrating Professor Themba A. Dube (A TAD Celebration I)
https://cgasa.sbu.ac.ir/article_104229.html
This 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&rsquo;s distinguished career and benefactions to the discipline with the early beginnings in nearness frames. We envelope the essential aspects of Dube&rsquo;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&rsquo;s imprints on certain categorical aspects of his work on &beta;L, &lambda;L, &upsilon;L and &szlig;L.Notes on the spatial part of a frame
https://cgasa.sbu.ac.ir/article_104138.html
A 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 &sube; L. First we analyze the behaviour of the spatial parts in the assembly: the points of L and of S(L)^op (&sim;=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 formulas$\alpha$-Projectable and laterally $\alpha$-complete Archimedean lattice-ordered groups with weak unit via topology
https://cgasa.sbu.ac.ir/article_104087.html
Let $\bf{W}$ 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 ($\omega &lt; \alpha$) $l(\alpha) C(X) = C(Yl(\alpha) C(X))$, then $X$ is finite. But ($\omega \leq \alpha$) for infinite $X$, $p(\alpha) C(X)$ sometimes is, and sometimes is not, $C(Yp(\alpha) C(X))$.S-Metrizability and the Wallman basis of a frame
https://cgasa.sbu.ac.ir/article_104094.html
The 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.A little more on ideals associated with sublocales
https://cgasa.sbu.ac.ir/article_104102.html
&nbsp; &nbsp; As usual, let $\mathcal RL$ denote the ring of real-valued continuous functions on a completely regular frame $L$. Let $\beta L$ and &nbsp;$\lambda L$ denote the &nbsp;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 &nbsp;of $\lambda L$ an ideal of $\mathcal RL$ in a manner similar to one of the ways one does &nbsp;it for sublocales of $\beta L$. &nbsp;The intent in this paper is to augment~\cite{DS1} by considering two other coreflections; namely, the realcompact and the paracompact &nbsp; coreflections.\\&nbsp; &nbsp; &nbsp; &nbsp; We show that $\boldsymbol M$-ideals of $\mathcal RL$ indexed by sublocales of $\beta L$ are precisely the intersections of maximal ideals of &nbsp;$\mathcal RL$. 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 &nbsp;$\boldsymbol{O}$-ideal of $\mathcal RL$. &nbsp;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 $\mathcal RL$.On one-local retract in modular metrics
https://cgasa.sbu.ac.ir/article_104146.html
We 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 $\lambda &gt;0$, has at least one fixed point whenever the collection of all $q_w$-admissible subsets of $X_{w}$ is both compact and normal.Direct products of cyclic semigroups and left zero semigroups in $\beta\mathbb{N}$
https://cgasa.sbu.ac.ir/article_104162.html
We 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 $\beta\mathbb{N}$.Topological spaces versus frames in the topos of $M$-sets
https://cgasa.sbu.ac.ir/article_104105.html
In 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 of topologies of points for frames, in our universe. Then we find functors between the categories of topological spaces and of frames in our universe.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. 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.Persian Abstracts for Vol. 20, No. 1.
https://cgasa.sbu.ac.ir/article_104228.html
Idempotent 2x2 matrices over linearly ordered abelian groups
https://cgasa.sbu.ac.ir/article_104001.html
In this paper we study multiplicative semigroups of $2\times 2$ matrices over a linearly ordered abelian group with an externally added bottom element. The multiplication of such a semigroup is defined by replacing addition and multiplication by join and addition in the usual formula defining matrix multiplication. We show that there are four types of idempotents in this semigroup and we determine which of them are $0$-primitive. We also prove that the poset of idempotents with respect to the natural order is a lattice. It turns out that this matrix semigroup is inverse or orthodox if and only if the abelian group is trivial.Combinatorial approach of the category $\Theta_0$ of cubical pasting diagrams
https://cgasa.sbu.ac.ir/article_104127.html
In globular higher category theory the small category $\Theta_0$ of finite rooted trees plays an important role: for example the objects of $\Theta_0$ are the arities of the operations inside the free globular $\omega$-operad $\mathbb{B}^0$ of Batanin, which $\mathbb{B}^0$-algebras are models of globular weak $\infty$-categories; also this globular $\Theta_0$ is an important tool to build the coherator $\Theta^{\infty}_{W^0}$ of Grothendieck which ${\mathbb{S}\text{ets}}$-models are globular weak $\infty$-groupoids. Cubical higher category needs similarly its $\Theta_0$. In this work we describe, combinatorially, the small category $\Theta_0$ which objects are cubical pasting diagrams and which morphisms are morphisms of cubical sets.&nbsp;The coherator $\Theta^{\infty}_W$ of cubical weak $\infty$-categories with connections
https://cgasa.sbu.ac.ir/article_104139.html
This work exhibits two applications of the combinatorial approach in [12] of the small category $\Theta_0$ which objects are cubical pasting diagrams. First we provide an accurate description of the monad $\mathbb{S}=(S,\lambda,\mu)$ acting on the category ${\mathbb{C}\mathbb{S}\text{ets}}$ of cubical sets (without degeneracies and connections), which algebras are cubical strict $\infty$-categories with connections, and show that this monad is cartesian, which solve a conjecture in \cite{camark-cub}. Secondly we give a precise construction of the cubical coherator $\Theta^{\infty}_W$ which set-models are cubical weak $\infty$-categories with connections, and we also give a precise construction of the cubical coherator $\Theta^{\infty}_{W^{0}}$ which set-models are cubical weak $\infty$-groupoids with connections. &nbsp;Characterization of monoids by ($U$-)$GPW$-flatness of right acts
https://cgasa.sbu.ac.ir/article_104288.html
The authors in 2020 introduced $GPW$-flatness and gave a characterization of monoids by this property of their right acts. In this article we continue this investigation and will give a characterization of monoids by this condition of their right Rees factor acts. Also we give a characterization of monoids by comparing this property of their &nbsp;right acts with other properties.We also introduce $U$-$GPW$-flatness of acts, which is an extension of $GPW$-flatness and give some general properties and a characterization of monoids when this property of acts implies some others and vice versa.&nbsp;Generalised geometric logic
https://cgasa.sbu.ac.ir/article_104391.html
This paper introduces a notion of generalised geometric logic. Connections of generalised geometric logic with the L-topological system and L-topological space are established.