Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Idempotent 2x2 matrices over linearly ordered abelian groups11710400110.48308/cgasa.2023.232266.1412ENValdisLaanInstitute of Mathematics and Statistics, University of Tartu, Tartu, EstoniaMarilynKuttiInstitute of Mathematics and Statistics, University of Tartu, Tartu, EstoniaJournal Article20230704In 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. <br />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.https://cgasa.sbu.ac.ir/article_104001_76293afda547ea032c46a02e3f20dbfe.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Combinatorial approach of the category $\Theta_0$ of cubical pasting diagrams196810412710.48308/cgasa.2023.104127ENCamellKachourLaboratoire de Math\'ematiques d'Orsay, UMR 8628,
Universit\'e de Paris-Saclay and CNRS,
B\^atiment 307, Facult\'e des Sciences d'Orsay,
94015 ORSAY Cedex, France.0009-0004-9550-1648Journal Article20231224In 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. https://cgasa.sbu.ac.ir/article_104127_62e974e23cde05d4e213607d89491bbf.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701The coherator $\Theta^{\infty}_W$ of cubical weak $\infty$-categories with connections6912610413910.48308/cgasa.2023.104139ENCamellKachourLaboratoire de Math\'ematiques d'Orsay, UMR 8628,
Universit\'e de Paris-Saclay and CNRS,
B\^atiment 307, Facult\'e des Sciences d'Orsay,
94015 ORSAY Cedex, France.0009-0004-9550-1648Journal Article20230914This 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. https://cgasa.sbu.ac.ir/article_104139_a20f622a319ee26057e1eac8347b7197.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Characterization of monoids by ($U$-)$GPW$-flatness of right acts12715210428810.48308/cgasa.2024.231706.1404ENHamidehRashidiDepartment of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran0000-0002-0496-040XAkbarGolchinDepartment of Mathematics, University of Sistan
and Baluchestan, Zahedan, IranHosseinMohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranJournal Article20230514The 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 right acts with other properties.<br />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. https://cgasa.sbu.ac.ir/article_104288_5d47fb88f41b96973ef5465d3dc4518d.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701δ-primary subhypermodules on Krasner hyperrings15317410456810.48308/cgasa.2024.234020.1446ENKostaqHilaDepartment of Mathematical Engineering, Polytechnic University of Tirana, Albania0000-0001-6425-2619ElifKayaDepartment of Mathematics and Science Education, Istanbul Sabahattin Zaim University, Istanbul, TurkiyeMelisBolatDepartment of Computer Engineering, Istanbul Gelisim University, Istanbul, TurkiyeBayram AliErsoyDepartment of Mathematics, Yildiz Technical University, Istanbul, TurkiyeSerkanOnarDepartment of Mathematical Engineering, Yildiz Technical University, Istanbul, TurkiyeBijanDavvazDepartment of Mathematics, Yazd University, Yazd, Iran0000-0003-1941-5372Journal Article20231204In 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.https://cgasa.sbu.ac.ir/article_104568_70a8d57a09a82a36c61acd840046028a.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Finitely presentable objects in ${\rm(}Cb\text{-}{\bf Sets}{\rm)}_{_{\rm fs}}$17520910461510.48308/cgasa.2024.235466.1487ENMahdiehHaddadiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.KhadijehKeshvardoostDepartment of Mathematics, Velayat University, Iranshahr, Sistan and
Balochistan, Iran.AliyehHosseinabadiFaculty of Mathematics, Statistics and Computer Sciences, Department of Mathematics, Semnan University, Semnan, Iran.Journal Article20240425Pitts 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. <br />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.<br />Then, in the category of finitely supported $Cb$-sets, we characterize the finitely presentable objects and provide a generator in this category.https://cgasa.sbu.ac.ir/article_104615_6c8f44a2a0ff2073e8f68a1546e5d917.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Classification of Boolean algebras through von Neumann regular $\mathcal{C}^{\infty}-$rings21123910472610.48308/cgasa.2024.104726ENJean CerqueiraBerniDepartment of Mathematics, S˜ao Paulo State University - UNESP,
13506-900 , S˜ao Paulo, Brazil.Hugo LuizMarianoo Department of Mathematics, University of S˜ao Paulo, 05508-090, S˜ao Paulo, Brazil.0000-0002-9745-2411Journal Article20240804In 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.https://cgasa.sbu.ac.ir/article_104726_0c3690a6bca4bd6bc3dabb0b5b956059.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585321120240701Bayer noise quasisymmetric functions and some combinatorial algebraic structures24128210466910.48308/cgasa.2024.233890.1442ENAdnan H.AbdulwahidCollege of Business, Engineering, and Technology, Texas A & M University--Texarkana, 7101, University Ave, Texarkana, TX, 75503, USA.Journal Article20231123Recently, 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 $\mathbf k$.https://cgasa.sbu.ac.ir/article_104669_acb341ede31321298cf1afcfbb5bb78d.pdf