فهرست مطالب

Categories and General Algebraic Structures with Applications - Volume:11 Issue: 2, 2020
  • Volume:11 Issue: 2, 2020
  • تاریخ انتشار: 1398/11/29
  • تعداد عناوین: 8
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  • Pawel Gladki*, Krzysztof Worytkiewicz Pages 1-23

    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.

    Keywords: Quadratically presentable fields, Witt rings, hyperfields, quadratic forms
  • Hamideh Rashidi*, Akbar Golchin, Hossein Mohammadzadeh Saany Pages 25-42

    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.

    Keywords: GPW-flat, Eventually regular monoid, Eventually left almost regular monoid
  • Ahsan Mahboob*, Noor Khan, Bijan Davvaz Pages 43-67

    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.

    Keywords: Ordered semihypergroups, $(m, 0)$-hyperideals, $(0, n)$-hyperideals
  • Najmeh Jafarzadeh, Reza Ameri * Pages 69-88

    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.

    Keywords: (m, n)-hypermodules, kernel, cokernel, balanced category, fundamental functor, exact category
  • Marco Grandis*, George Janelidze Pages 89-121

    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].

    Keywords: Exact sequence, torsion theory, closure operator, factorization system, ideal of null morphisms
  • Alireza Ahmadi, Akbar Dehghan Nezhad* Pages 123-147

    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.

    Keywords: Cosheaves, quasi-cosheaves, site of plots, covering generating families, quasi-v{C}ech homology, diffeological spaces
  • Muhammad Qasim*, Samed &Ouml Zkan Pages 149-173

    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.

    Keywords: $mathcal{L}$-approach distance space, $mathcal{L}$-gauge space, topological category, Separation, closedness, D-connectedness
  • Pouyan Khamechi, Hossein Mohammadzadeh Saany, Leila Nouri* Pages 175-197

    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}).

    Keywords: S-act, Flatness properties, Condition (PWP, {ssc}), semi-cancellative, e-cancellative