Categories and General Algebraic Structures with Applications
https://cgasa.sbu.ac.ir/
Categories and General Algebraic Structures with Applicationsendaily1Fri, 01 Jan 2021 00:00:00 +0330Fri, 01 Jan 2021 00:00:00 +0330Cover for Vol. 14, No. 1.
https://cgasa.sbu.ac.ir/article_100670.html
Cofree objects in the centralizer and the center categories
https://cgasa.sbu.ac.ir/article_100669.html
We study cocompleteness, co-wellpoweredness, and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness, and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.On general closure operators and quasi factorization structures
https://cgasa.sbu.ac.ir/article_87435.html
In this article the notions of quasi mono (epi) as a generalization of mono (epi), (quasi weakly hereditary) general closure operator $\mathbf{C}$ on a category $\mathcal{X}$ with respect to a class $\mathcal{M}$ of morphisms, and quasi factorization structures in a category $\mathcal{X}$ are introduced. It is shown that under certain conditions, if $(\mathcal{E}, \mathcal{M})$ is a quasi factorization structure in $\mathcal{X}$, then $\mathcal{X}$ has a quasi right $\mathcal{M}$-factorization structure and a quasi left $\mathcal{E}$-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class $\mathcal{M}$, every quasi factorization structure $(\mathcal{E}, \mathcal{M})$ yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class $\mathcal{M}$, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. Several illustrative examples are provided.Duality theory of $p$-adic Hopf algebras
https://cgasa.sbu.ac.ir/article_87523.html
We show the monoidal functoriality of Schikhof duality, and cultivate new duality theory of $p$-adic Hopf algebras. Through the duality, we introduce two sorts of $p$-adic Pontryagin dualities. One is a duality between discrete Abelian groups and affine formal group schemes of specific type, and the other one is a duality between profinite Abelian groups and analytic groups of specific type. We extend Amice transform to a $p$-adic Fourier transform compatible with the second $p$-adic Pontryagin duality. As applications, we give explicit presentations of a universal family of irreducible $p$-adic unitary Banach representations of the open unit disc of the general linear group and its $q$-deformation in the case of dimension $2$.Schneider-Teitelbaum duality for locally profinite groups
https://cgasa.sbu.ac.ir/article_87524.html
We define monoidal structures on several categories of linear topological modules over the valuation ring of a local field, and study module theory with respect to the monoidal structures. We extend the notion of the Iwasawa algebra to a locally profinite group as a monoid with respect to one of the monoidal structure, which does not necessarily form a topological algebra. This is one of the main reasons why we need monoidal structures. We extend Schneider--Teitelbaum duality to duality applicable to a locally profinite group through the module theory over the generalised Iwasawa algebra, and give a criterion of the irreducibility of a unitary Banach representation.Constructing the Banaschewski compactification through the functionally countable subalgebra of $C(X)$
https://cgasa.sbu.ac.ir/article_87513.html
Let $X$ be a zero-dimensional space and $C_c(X)$ denote the functionally countable subalgebra of $C(X)$. It is well known that $\beta_0X$ (the Banaschewski compactfication of $X$) is a quotient space of $\beta X$. In this article, we investigate a construction of $\beta_0X$ via $\beta X$ by using $C_c(X)$ which determines the quotient space of $\beta X$ homeomorphic to&nbsp; $\beta_0X$. Moreover, the construction of&nbsp; $\upsilon_0X$ via $\upsilon_{_{C_c}}X$ (the subspace&nbsp; $\{p\in \beta X: \forall f\in C_c(X), f^*(p)&lt;\infty\}$ of $\beta X$) is also investigated.On bornological semi-abelian algebras
https://cgasa.sbu.ac.ir/article_87514.html
If $\Bbb T$ is a semi-abelian algebraic theory, we prove that the category ${\rm Born}^{\Bbb T}$ of bornological $\Bbb T$-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in ${\rm Born}^{\Bbb T}$ and, for a bornological $\Bbb T$-algebra $X$, we investigate the relation between the representability of actions on $X$ as a $\Bbb T$-algebra and as a bornological $\Bbb T$-algebra. We investigate further the algebraic coherence and the algebraic local cartesian closedness of ${\rm Born}^{\Bbb T}$ and prove in particular that both properties hold in the case of bornological groups.Distributive lattices and some related topologies in comparison with zero-divisor graphs
https://cgasa.sbu.ac.ir/article_94188.html
In this paper,for a distributive lattice $\mathcal L$, we study and compare some lattice theoretic features of $\mathcal L$ and topological properties of the Stone spaces ${\rm Spec}(\mathcal L)$ and ${\rm Max}(\mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $\Gamma(\mathcal L)$.Among other things,we show that the Goldie dimension of $\mathcal L$ is equal to the cellularity of the topological space ${\rm Spec}(\mathcal L)$ which is also equal to the clique number of the zero-divisor graph $\Gamma(\mathcal L)$. Moreover, the domination number of $\Gamma(\mathcal L)$ will be compared with the density and the weight of the topological space ${\rm Spec}(\mathcal L)$.&nbsp;For a $0$-distributive lattice $\mathcal L$, we investigate the compressed subgraph $\Gamma_E(\mathcal L)$ of the zero-divisor graph $\Gamma(\mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $\mathcal L$.Relation between Sheffer Stroke and Hilbert algebras
https://cgasa.sbu.ac.ir/article_87510.html
In this paper, we introduce a Sheffer stroke Hilbert algebra by giving definitions of Sheffer stroke and a Hilbert algebra. After it is shown that the axioms of Sheffer stroke Hilbert algebra are independent, it is given some properties of this algebraic structure. Then it is stated the relationship between Sheffer stroke Hilbert algebra and Hilbert algebra by defining a unary operation on Sheffer stroke Hilbert algebra. Also, it is presented deductive system and ideal of this algebraic structure. It is defined an ideal generated by a subset of a Sheffer stroke Hilbert algebra, and it is constructed a new ideal of this algebra by adding an element of this algebra to its ideal.Persian Abstracts, Vol. 14, No. 1.
https://cgasa.sbu.ac.ir/article_100671.html
Abundant Semigroups With Medial Idempotents
https://cgasa.sbu.ac.ir/article_87496.html
The effect of the existence of a medial or related idempotent in any abundant semigroup is the subject of this paper. The aim is to naturally order any abundant semigroup $S$ which contains an ample multiplicative medial idempotent $u$ in a way that $\mathcal{L}^*$ and $\mathcal{R}^*$ are compatible&nbsp; with the natural order and $u$ is a maximum idempotent. The structure of an abundant semigroup containing an ample normal medial idempotent studied in \cite{item6} will be revisited.Pre-image of functions in $C(L)$
https://cgasa.sbu.ac.ir/article_100691.html
Let $C(L)$ be the ring of all continuous real functions on a frame $L$ and $S\subseteq{\mathbb R}$. An $\alpha\in C(L)$ is said to be an overlap of $S$, denoted by $\alpha\blacktriangleleft S$, whenever $u\cap S\subseteq v\cap S$ implies $\alpha(u)\leq\alpha(v)$ for every open sets $u$ and $v$ in $\mathbb{R}$. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in {\it Pointfree version of image of real-valued continuous functions} (2018). Although this concept is a suitable model for their purpose, it ultimately does not provide a clear definition of the range of continuous functions in the context of pointfree topology. In this paper, we will introduce a concept which is called pre-image, denoted by ${\rm pim}$, as a pointfree version of the image of real-valued continuous functions on a topological space $X$. We investigate this concept and in addition to showing ${\rm pim}(\alpha)=\bigcap\{S\subseteq{\mathbb R}:~\alpha\blacktriangleleft S\}$, we will see that this concept is a good surrogate for the image of continuous real functions. For instance, we prove, under some achievable conditions, we have ${\rm pim}(\alpha\vee\beta)\subseteq {\rm pim}(\alpha)\cup {\rm pim}(\beta)$, ${\rm pim}(\alpha\wedge\beta)\subseteq {\rm pim}(\alpha)\cap {\rm pim}(\beta)$, ${\rm pim}(\alpha\beta)\subseteq {\rm pim}(\alpha){\rm pim}(\beta)$ and ${\rm pim}(\alpha+\beta)\subseteq {\rm pim}(\alpha)+{\rm pim}(\beta)$.Flatness properties of acts over semigroups
https://cgasa.sbu.ac.ir/article_101062.html
In this paper we study flatness properties (pullback flatness, limit flatness, finite limit flatness) of acts over semigroups. These are defined by requiring preservation of certain limits from the functor of tensor multiplication by a given act. We give a description of firm pullback flat acts using Conditions (P) and (E). We also study pure epimorphisms and their connections to finitely presented acts and pullback flat acts. We study these flatness properties in the category of all acts, as well as in the category of unitary acts and in the category of firm acts, which arise naturally in the Morita theory of semigroups.&nbsp;Simplicial Structures Over the 3-sphere and Generalized Higher Order Hochschild Homology
https://cgasa.sbu.ac.ir/article_101063.html
In this paper, we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple $(A,B,C,\varepsilon,\theta)$, which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations.Six model categories for directed homotopy
https://cgasa.sbu.ac.ir/article_101064.html
We construct a q-model structure, an h-model structure and an m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and left determined and they coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.The elementary construction of formal anafunctors
https://cgasa.sbu.ac.ir/article_101133.html
This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully faithful.On epimorphisms and structurally regular semigroups
https://cgasa.sbu.ac.ir/article_101189.html
In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n,m)-regular semigroups for any pair (n,m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n,m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n,m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimorphisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1 &le; k &le; n.(b, c)-inverse, inverse along an element, and the Schützenberger category of a semigroup
https://cgasa.sbu.ac.ir/article_101196.html
We prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Sch&uuml;tzenberger category of a semigroup. Applications to the Reverse Order Law are given.