Categories and General Algebraic Structures with Applications
https://cgasa.sbu.ac.ir/
Categories and General Algebraic Structures with Applicationsendaily1Mon, 01 Jul 2024 00:00:00 +0330Mon, 01 Jul 2024 00:00:00 +0330Cover for Vol. 21, No. 1.
https://cgasa.sbu.ac.ir/article_104724.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;δ-primary subhypermodules on Krasner hyperrings
https://cgasa.sbu.ac.ir/article_104568.html
In this paper, we study commutative Krasner hyperrings with nonzero identity and nonzero unital hypermodules. We introduce a new concept, the $\delta$-primary subhypermodule on Krasner hyperrings. Some characterizations and properties for $\delta$-primary subhypermodules using the expansion function $\delta$ are provided. The images and inverse images of $\delta$-primary subhypermodules under homomorphism are investigated. Finally, some characterizations for multiplication hypermodules with some special conditions are provided.Finitely presentable objects in ${\rm(}Cb\text{-}{\bf Sets}{\rm)}_{_{\rm fs}}$
https://cgasa.sbu.ac.ir/article_104615.html
Pitts generalized nominal sets to finitely supported $Cb$-sets by utilizing the monoid $Cb$ of name substitutions instead of the monoid of finitary permutations over names. Finitely supported $Cb$-sets provide a framework for studying essential ideas of models of homotopy type theory at the level of convenient abstract categories. &nbsp;&nbsp;Here, the interplay of two separate categories of finitely supported actions of a submonoid of ${\rm End}(\mathbb {D})$, for some countably infinite set $\mathbb {D}$, over sets is first investigated. In particular, we specify the structure of free objects.Then, in the category of finitely supported $Cb$-sets, we characterize the finitely presentable objects and provide a generator in this category.Classification of Boolean algebras through von Neumann regular $\mathcal{C}^{\infty}-$rings
https://cgasa.sbu.ac.ir/article_104726.html
In this paper, we introduce the concept of a ``von Neumann regular $\mathcal{C}^{\infty}$-ring", which is a model for a specific equational theory. We delve into the characteristics of these rings and demonstrate that each Boolean space can be effectively represented as the image of a von Neumann regular $\mathcal{C}^{\infty}$-ring through a specific functor. Additionally, we establish that every homomorphism between Boolean algebras can be expressed through a $\mathcal{C}^{\infty}$-ring homomorphism between von Neumann regular $\mathcal{C}^{\infty}$-rings.Bayer noise quasisymmetric functions and some combinatorial algebraic structures
https://cgasa.sbu.ac.ir/article_104669.html
Recently, quasisymmetric functions have been widely studied due to their big connection to enumerative combinatorics, combinatorial Hopf algebra and number theory. The Bayer filter mosaic, named due to Bryce Bayer (1929-2012), is a color filter array used to arrange RGB color filters on a square grid of photosensors. It is the most common pattern of filters, and almost all professional digital cameras are applications of this filter. We use this filter to introduce the Bayer Noise quasisymmetric functions, and we study some combinatorial algebraic and coalgebraic structures on Quasi-Bayer Noise modules and on Quasi-Bayer GB-Noise modules. We explicitly describe the primitive basis elements for each comultiplication defined on Quasi-Bayer Noise modules, and we calculate different kinds of comultiplications defined on Quasi-Bayer Noises module over a fixed commutative ring $\mathbf k$.Persian Abstracts for Vol. 21, No. 1.
https://cgasa.sbu.ac.ir/article_104725.html
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.Another closure operator on preneighbourhood spaces
https://cgasa.sbu.ac.ir/article_104597.html
The notions of dense, proper, separated or perfect morphisms and hence of compact, Hausdorff or compact Hausdorff are all consequent to good properties of a family of closed morphisms is well known in literature. Deeper consequences like the Tychonoff product theorem or the Stone Čech compactifications follow from richer properties of the set of closed morphisms. The purpose of this paper is to provide a closure operation on a preneighbourhood space so that the resulting set of closed morphisms possess all the properties mentioned above.&nbsp;A correspondence between proximity homomorphisms and certain frame maps via a comonad
https://cgasa.sbu.ac.ir/article_104668.html
We exhibit the proximity frames and proximity homomorphisms as a Kleisli category of a comonad whose underlying functor takes a proximity frame &nbsp;to its frame of round ideals. This construction is known in the literature as stable compactification ([6]). We show that the frame of round ideals naturally carries with it two proximities of interest from which two comonads are induced.&nbsp;Baer criterion in locally presentable categories
https://cgasa.sbu.ac.ir/article_104674.html
In this paper, some Baer type criteria are considered for locally presentable categories. Recalling the notion of the classical Baer criterion for injectivity, it is shown that a locally presentable category which has enough injectives and coproduct injections, which are monomorphisms, satisfy such criterion if and only if the class of its injective objects is accessibly embedded in the category. Also, it is shown that this criterion is equivalent to the Baer type criterion that injectivity is equivalent to injectivity with respect to a subclass of monomorphisms.It is also proved some Baer type criteria for $\lambda$-presentable categories for injectivity with respect to monomorphisms with $\lambda$-presentable domains and codomains, for a regular cardinal number $\lambda$.In particular, some Baer type criteria is found for varieties.Inductive graded rings, hyperfields and quadratic forms
https://cgasa.sbu.ac.ir/article_104696.html
In [6] we developed a k-theory for the category of hyperbolic hyperfields (a category that contains a copy of the category of (pre)special groups): this construction extends, simultaneously, Milnor's k-theory ([20]) and Dickmann-Miraglia's k-theory ([13]). An abstract environment that encapsulate all them, and of course, provide an axiomatic approach to guide new extensions of the concept of K-theory in the context of the algebraic and abstract theories of quadratic forms is given by the concept of inductive graded rings a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in realm the algebraic theory of quadratic forms for Pythagorean fields. The goal of this work is twofold: (i) to provide a detailed analysis of some categories of inductive graded ring - a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in the algebraic theory of quadratic forms; (ii) apply this analysis to deepen the connections between the category of special hyperfields ([6]) - equivalent to the category of special groups ([10]) and the categories of inductive graded rings.