CATEGORIES AND GENERAL ALGEBRAIC STRUCTURES WITH APPLICATIONS
Year:2020 | Volume:12 | Issue:1 |Papers Count:8

This 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.

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 AS is GPW-flat if for every s 2 S, there exists a natural number n = n(s, AS) 2 N such that the functor AS S− preserves the embedding of the principal left ideal S(Ssn) into SS. We show that a right S-act AS is GPW-flat if and only if for every s 2 S there exists a natural number n = n(s, AS) 2 N such that the corresponding ' is surjective for the pullback diagram P(Ssn, Ssn,  ,  , S), where  : S(Ssn)! SS 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.

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 mI, In, Hnm, and Bnm on an ordered semihypergroup are, then, introduced. We prove that Bnm  Hnm 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 Hnm-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 Bnm-class and Hnm-class containing that element, respectively.

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.

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].

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 Poincaré 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 quasicosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-Cech homology with values in pre-cosheaves is established for diffeological spaces.

In this paper, we characterize local T0 and T1 quantale-valued gauge spaces, show how these concepts are related to each other and apply them to L-approach distance spaces and 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.

Condition (PWP) which was introduced by Laan is related to the concept of flatness of acts over monoids. Golchin and Mohammadzadeh introduced Condition (PWPE) as a generalization of Condition (PWP). In this paper, we introduce Condition (PWPssc) which is much easier to check than Conditions (PWP) and (PWPE) and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition (PWPssc) will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition (PWPssc) and also a classification of monoids S will be given for which all right S-acts satisfying some other flatness properties have Condition (PWPssc).