Categories and General Algebraic Structures with ApplicationsCategories and General Algebraic Structures with Applications
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Tue, 07 Jul 2020 08:44:26 +0100FeedCreatorCategories and General Algebraic Structures with Applications
http://cgasa.sbu.ac.ir/
Feed provided by Categories and General Algebraic Structures with Applications. Click to visit.Cover for Vol. 12, No. 1.
http://cgasa.sbu.ac.ir/article_87446_11662.html
Tue, 31 Dec 2019 20:30:00 +0100Witt rings of quadratically presentable fields
http://cgasa.sbu.ac.ir/article_87412_11662.html
This paper introduces an approach to the axiomatic theory of quadratic forms based on {tmem{presentable}} partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of {tmem{quadratically presentable fields}}, that is, fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. In particular, Witt rings of symmetric bilinear forms over fields of arbitrary characteristics are isomorphic to Witt rings of suitably built quadratically presentable fields.Tue, 31 Dec 2019 20:30:00 +0100On $GPW$-Flat Acts
http://cgasa.sbu.ac.ir/article_82637_11662.html
In this article, we present $GPW$-flatness property of acts over monoids, which is a generalization of principal weak flatness. We say that a right $S$-act $A_{S}$ is $GPW$-flat if for every $s in S$, there exists a natural number $n = n_ {(s, A_{S})} in mathbb{N}$ such that the functor $A_{S} otimes {}_{S}- $ preserves the embedding of the principal left ideal ${}_{S}(Ss^n)$ into ${}_{S}S$. We show that a right $S$-act $A_{S}$ is $GPW$-flat if and only if for every $s in S$ there exists a natural number $n = n_{(s, A_{S})} in mathbb{N}$ such that the corresponding $varphi$ is surjective for the pullback diagram $P(Ss^n, Ss^n, iota, iota, S)$, where $iota : {}_{S}(Ss^n) rightarrow {}_{S}S$ is a monomorphism of left $S$-acts. Also we give some general properties and a characterization of monoids for which this condition of their acts implies some other properties and vice versa.Tue, 31 Dec 2019 20:30:00 +0100$(m,n)$-Hyperideals in Ordered Semihypergroups
http://cgasa.sbu.ac.ir/article_87415_11662.html
In this paper, first we introduce the notions of an $(m,n)$-hyperideal and a generalized $(m,n)$-hyperideal in an ordered semihypergroup, and then, some properties of these hyperideals are studied. Thereafter, we characterize $(m,n)$-regularity, $(m,0)$-regularity, and $(0,n)$-regularity of an ordered semihypergroup in terms of its $(m,n)$-hyperideals, $(m,0)$-hyperideals and $(0,n)$-hyperideals, respectively. The relations ${_mmathcal{I}}, mathcal{I}_n, mathcal{H}_m^n$, and $mathcal{B}_m^n$ on an ordered semihypergroup are, then, introduced. We prove that $mathcal{B}_m^n subseteq mathcal{H}_m^n$ on an ordered semihypergroup and provide a condition under which equality holds in the above inclusion. We also show that the $(m,0)$-regularity [$(0,n)$-regularity] of an element induce the $(m,0)$-regularity [$(0,n)$-regularity] of the whole $mathcal{H}_m^n$-class containing that element as well as the fact that $(m,n)$-regularity and $(m,n)$-right weakly regularity of an element induce the $(m,n)$-regularity and $(m,n)$-right weakly regularity of the whole $mathcal{B}_m^n$-class and $mathcal{H}_m^n$-class containing that element, respectively.Tue, 31 Dec 2019 20:30:00 +0100On exact category of $(m, n)$-ary hypermodules
http://cgasa.sbu.ac.ir/article_80792_11662.html
We introduce and study category of $(m, n)$-ary hypermodules as a generalization of the category of $(m, n)$-modules as well as the category of classical modules. Also, we study various kinds of morphisms. Especially, we characterize monomorphisms and epimorphisms in this category. We will proceed to study the fundamental relation on $(m, n)$-hypermodules, as an important tool in the study of algebraic hyperstructures and prove that this relation is really functorial, that is, we introduce the fundamental functor from the category of $(m, n)$-hypermodules to the category $(m, n)$-modules and prove that it preserves monomorphisms. Finally, we prove that the category of $(m, n)$-hypermodules is an exact category, and, hence, it generalizes the classical case.Tue, 31 Dec 2019 20:30:00 +0100From torsion theories to closure operators and factorization systems
http://cgasa.sbu.ac.ir/article_87116_11662.html
Torsion theories are here extended to categories equipped with an ideal of 'null morphisms', or equivalently a full subcategory of 'null objects'. Instances of this extension include closure operators viewed as generalised torsion theories in a 'category of pairs', and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15].Tue, 31 Dec 2019 20:30:00 +0100Some aspects of cosheaves on diffeological spaces
http://cgasa.sbu.ac.ir/article_87119_11662.html
We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of the D-topological structure. We also study quasi-cosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-v{C}ech homology with values in pre-cosheaves is established for diffeological spaces.Tue, 31 Dec 2019 20:30:00 +0100The notions of closedness and D-connectedness in quantale-valued approach spaces
http://cgasa.sbu.ac.ir/article_87411_11662.html
In this paper, we characterize local $T_{0}$ and $T_{1}$ quantale-valued gauge spaces, show how these concepts are related to each other and apply them to $mathcal{L}$-approach distance spaces and $mathcal{L}$-approach system spaces. Furthermore, we give the characterization of a closed point and $D$-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each other.Tue, 31 Dec 2019 20:30:00 +0100Classification of monoids by Condition $(PWP_{ssc})$
http://cgasa.sbu.ac.ir/article_85729_11662.html
Condition $(PWP)$ which was introduced in (Laan, V., {it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $(PWP_E)$ is a generalization of Condition $(PWP)$. In this paper we introduce Condition $(PWP_{ssc})$, which is much easier to check than Conditions $(PWP)$ and $(PWP_E)$ and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition $(PWP_{ssc})$ will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition $(PWP_{ssc})$ and also a classification of monoids $S$ will be given for which all right $S$-acts satisfying some other flatness properties have Condition $(PWP_{ssc})$.Tue, 31 Dec 2019 20:30:00 +0100Persian Abstracts, Vol. 11, No. 1.
http://cgasa.sbu.ac.ir/article_87447_11662.html
Tue, 31 Dec 2019 20:30:00 +0100Separated finitely supported $Cb$-sets
http://cgasa.sbu.ac.ir/article_87413_0.html
The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$-set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$-set, there exists an element of the monoid $Cb$ which separates them by making just one of them into an element with the empty support.In this paper, we generalize these properties of simple finitely supported $Cb$-sets by modifying slightly the notion of the support map; defining the notion of $mathsf{2}$-equivariant support map; and introducing the notions of s-separated and z-separated finitely supported $Cb$-sets. We show that the notions of s-separated and z-separated coincide for a finitely supported $Cb$-set whose support map is $mathsf{2}$-equivariant. Among other results, we find a characterization of simple s-separated (or z-separated) finitely supported $Cb$-sets. Finally, we show that some subcategories of finitely supported $Cb$-sets with injective equivariant maps which constructed applying the defined notions are reflective.Wed, 27 Nov 2019 20:30:00 +0100Product preservation and stable units for reflections into idempotent subvarieties
http://cgasa.sbu.ac.ir/article_87414_0.html
We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the above-mentioned condition holds.Wed, 27 Nov 2019 20:30:00 +0100The symmetric monoidal closed category of cpo $M$-sets
http://cgasa.sbu.ac.ir/article_87434_0.html
In this paper, we show that the category of directed complete posets with bottom elements (cpos) endowed with an action of a monoid $M$ on them forms a monoidal category. It is also proved that this category is symmetric closed.Sun, 22 Dec 2019 20:30:00 +0100On general closure operators and quasi factorization structures
http://cgasa.sbu.ac.ir/article_87435_0.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.Sun, 22 Dec 2019 20:30:00 +0100The non-abelian tensor product of normal crossed submodules of groups
http://cgasa.sbu.ac.ir/article_87437_0.html
In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.Tue, 31 Dec 2019 20:30:00 +0100Tense like equality algebras
http://cgasa.sbu.ac.ir/article_87465_0.html
In this paper, first we define the notion of involutive operator on bounded involutive equality algebras and by using it, we introduce a new class of equality algebras that we called it a tense like equality algebra. Then we investigate some properties of tense like equality algebra. For two involutive bounded equality algebras and an equality homomorphism between them, we prove that the tense like equality algebra structure can be transfer by this equality homomorphism. Specially, by using a bounded involutive equality algebra and quotient structure of it, we construct a quotient tense like equality algebra. Finally, we investigate the relation between tense like equality algebras and tense MV-algebras.Fri, 14 Feb 2020 20:30:00 +0100Abundant Semigroups With Medial Idempotents
http://cgasa.sbu.ac.ir/article_87496_0.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 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.Thu, 14 May 2020 19:30:00 +0100Relation between Sheffer Stroke and Hilbert algebras
http://cgasa.sbu.ac.ir/article_87510_0.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.Sat, 06 Jun 2020 19:30:00 +0100Crossed squares, crossed modules over groupoids and cat$^{\bf {1-2}}-$groupoids
http://cgasa.sbu.ac.ir/article_87511_0.html
The aim of this paper is to introduce the notion of cat$^{bf {1}}-$groupoids which are the groupoid version of cat$^{bf {1}}-$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{bf {1}}-$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{bf {2}}-$groupoids, and then we show their categories are equivalent. These equivalences enable us to obtain more examples of groupoids.Sat, 06 Jun 2020 19:30:00 +0100Distributive lattices with strong endomorphism kernel property as direct sums
http://cgasa.sbu.ac.ir/article_87512_0.html
Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem 2.8}). We shall determine the structure of special elements (which are introduced after Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of three lattices, a lattice with exactly one strong element, a lattice which is a direct sum of 2 element lattices with distinguished elements 1 and a lattice which is a direct sum of 2 element lattices with distinguished elements 0, and the sublattice of strong elements is isomorphic to a product of last two mentioned lattices.Sat, 06 Jun 2020 19:30:00 +0100Constructing the Banaschewski compactification through the functionally countable subalgebra of ...
http://cgasa.sbu.ac.ir/article_87513_0.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 $beta_0X$. Moreover, the construction of $upsilon_0X$ via $upsilon_{_{C_c}}X$ (the subspace ${pin beta X: forall fin C_c(X), f^*(p)<infty}$ of $beta X$) is also investigated.Sat, 06 Jun 2020 19:30:00 +0100On bornological semi-abelian algebras
http://cgasa.sbu.ac.ir/article_87514_0.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.Sat, 06 Jun 2020 19:30:00 +0100Duality theory of $p$-adic Hopf algebras
http://cgasa.sbu.ac.ir/article_87523_0.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 $2$.Thu, 25 Jun 2020 19:30:00 +0100Schneider-Teitelbaum duality for locally profinite groups
http://cgasa.sbu.ac.ir/article_87524_0.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.Thu, 25 Jun 2020 19:30:00 +0100