Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101Witt rings of quadratically presentable fields1238741210.29252/cgasa.12.1.1ENPawelGladkiInstitute of Mathematics, Faculty of Mathematics, Physics and Chemistry, University of SilesiaKrzysztofWorytkiewiczLaboratorire de Math'{e}matiques, Universit'{e} Savoie Mont Blanc, B^{a}timent Le Chablais, Campus Scientifique, 73376 Le Bourget du Lac, France.Journal Article20180426This paper introduces an approach to the axiomatic theory of quadratic forms based on $presentable$ 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 $quadratically presentable fields$, 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101On $GPW$-Flat Acts25428263710.29252/cgasa.12.1.25ENHamidehRashidiDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.AkbarGolchinUniversity of Sistan and BaluchestanHosseinMohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranJournal Article20180424In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101$(m,n)$-Hyperideals in Ordered Semihypergroups43678741510.29252/cgasa.12.1.43ENAhsanMahboobAligarh Muslim UniversityNoor MohammadKhanAligarh Muslim UniversityBijanDavvazYazd UniversityJournal Article20190405In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101On exact category of $(m, n)$-ary hypermodules69888079210.29252/cgasa.12.1.69ENNajmehJafarzadehDepartment of Mathematics, Payamenoor University,P.O. Box 19395-3697, Tehran, Iran.RezaAmeriMathematics, School of Mathematics, Statistics and Computer
Science, University of Tehran0000-0001-5760-1788Journal Article20170612We 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101From torsion theories to closure operators and factorization systems891218711610.29252/cgasa.12.1.89ENMarcoGrandisDipartimento di Matematica, Universit\`a di Genova, Via Dodecaneso 35,
16146-Genova, ItalyGeorgeJanelidzeDepartment of Mathematics and Applied Mathematics, University of Cape Town, South Africa.Journal Article20180510Torsion 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].Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101Some aspects of cosheaves on diffeological spaces1231478711910.29252/cgasa.12.1.123ENAlireza AlirezaAhmadiDepartment of Math. Yazd University
Yazd, IranAkbarDehghan NezhadSchool of Mathematics, Iran University of Science and Technology,
Narmak,Tehran, 16846--13114, IranJournal Article20181017We 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101The notions of closedness and D-connectedness in quantale-valued approach spaces1491738741110.29252/cgasa.12.1.149ENMuhammadQasimDepartment of Mathematics, School of Natural Sciences, National University of Sciences & Technology, Islamabad.0000-0001-9485-8072SamedOzkanDepartment of Mathematics, Hacı Bektaş Veli University, Nevşehir, TurkeyJournal Article20181225In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585312120200101Classification of monoids by Condition $(PWP_{ssc})$1751978572910.29252/cgasa.12.1.175ENPouyanKhamechiDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranHosseinMohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranLeilaNouriDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranJournal Article20180224Condition $(PWP)$ 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 $(PWP)$. <br /><br />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 $(PWP_{ssc})$.