Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
Witt rings of quadratically presentable fields
1
23
EN
Pawel
Gladki
Institute of Mathematics, Faculty of Mathematics, Physics and Chemistry, University of Silesia
pawel.gladki@us.edu.pl
Krzysztof
Worytkiewicz
Laboratorire de Math'{e}matiques, Universit'{e} Savoie Mont Blanc, B^{a}timent Le Chablais, Campus Scientifique, 73376 Le Bourget du Lac, France.
krzysztof.worytkiewicz@univ-smb.fr
10.29252/cgasa.12.1.1
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.
Quadratically presentable fields,Witt rings,hyperfields,quadratic forms
https://cgasa.sbu.ac.ir/article_87412.html
https://cgasa.sbu.ac.ir/article_87412_e4ca569b071e83128b5db22ac6d06101.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
On $GPW$-Flat Acts
25
42
EN
Hamideh
Rashidi
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.
hrashidi@pgs.usb.ac.ir
Akbar
Golchin
University of Sistan and Baluchestan
agdm@math.usb.ac.ir
Hossein
Mohammadzadeh Saany
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
hmsdm@math.usb.ac.ir
10.29252/cgasa.12.1.25
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.
$GPW$-flat,Eventually regular monoid,Eventually left almost regular monoid
https://cgasa.sbu.ac.ir/article_82637.html
https://cgasa.sbu.ac.ir/article_82637_db225e4212ba0171013678302be2c9d2.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
$(m,n)$-Hyperideals in Ordered Semihypergroups
43
67
EN
Ahsan
Mahboob
Aligarh Muslim University
khanahsan56@gmail.com
Noor
Mohammad
Khan
Aligarh Muslim University
nm_khan123@yahoo.co.in
Bijan
Davvaz
Yazd University
davvaz@yazd.ac.ir
10.29252/cgasa.12.1.43
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.
Ordered semihypergroups,$(m,0)$-hyperideals,$(0,n)$-hyperideals
https://cgasa.sbu.ac.ir/article_87415.html
https://cgasa.sbu.ac.ir/article_87415_1fd525cccd124d58a33309087242f95f.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
On exact category of $(m, n)$-ary hypermodules
69
88
EN
Najmeh
Jafarzadeh
Department of Mathematics, Payamenoor University,P.O. Box 19395-3697, Tehran, Iran.
jafarzadeh@phd.pnu.ac.ir
Reza
Ameri
0000-0001-5760-1788
Mathematics, School of Mathematics, Statistics and Computer
Science, University of Tehran
rameri@ut.ac.ir
10.29252/cgasa.12.1.69
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.
$(m,n)$-hypermodules,kernel,cokernel,balanced category,fundamental functor,exact category
https://cgasa.sbu.ac.ir/article_80792.html
https://cgasa.sbu.ac.ir/article_80792_907e526521584c03372aaada0e600e45.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
From torsion theories to closure operators and factorization systems
89
121
EN
Marco
Grandis
Dipartimento di Matematica, Universit\`a di Genova, Via Dodecaneso 35,
16146-Genova, Italy
grandis@dima.unige.it
George
Janelidze
Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa.
george.janelidze@uct.ac.za
10.29252/cgasa.12.1.89
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].
Exact sequence,torsion theory,closure operator,factorization system,ideal of null morphisms
https://cgasa.sbu.ac.ir/article_87116.html
https://cgasa.sbu.ac.ir/article_87116_929764e335e7d4c92a5611139b9e065a.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
Some aspects of cosheaves on diffeological spaces
123
147
EN
Alireza
Alireza
Ahmadi
Department of Math. Yazd University
Yazd, Iran
ahmadi@stu.yazd.ac.ir
Akbar
Dehghan Nezhad
School of Mathematics, Iran University of Science and Technology,
Narmak,Tehran, 16846--13114, Iran
dehghannezhad@iust.ac.ir
10.29252/cgasa.12.1.123
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.
Cosheaves,quasi-cosheaves,site of plots,covering generating families,quasi-v{C}ech homology,diffeological spaces
https://cgasa.sbu.ac.ir/article_87119.html
https://cgasa.sbu.ac.ir/article_87119_6b625005860bfe6a4bd5da17a099b89b.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
The notions of closedness and D-connectedness in quantale-valued approach spaces
149
173
EN
Muhammad
Qasim
0000-0001-9485-8072
Department of Mathematics, School of Natural Sciences, National University of Sciences & Technology, Islamabad.
muhammad.qasim@sns.nust.edu.pk
Samed
Ozkan
Department of Mathematics, Hacı Bektaş Veli University, Nevşehir, Turkey
ozkans@nevsehir.edu.tr
10.29252/cgasa.12.1.149
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.
$mathcal{L}$-approach distance space,$mathcal{L}$-gauge space,topological category,Separation,closedness,D-connectedness
https://cgasa.sbu.ac.ir/article_87411.html
https://cgasa.sbu.ac.ir/article_87411_0927cf623a93d8d592cac3c1677607b0.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
12
1
2020
01
01
Classification of monoids by Condition $(PWP_{ssc})$
175
197
EN
Pouyan
Khamechi
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
pouyan_khamechi@pgs.usb.ac.ir
Hossein
Mohammadzadeh Saany
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
hmsdm@math.usb.ac.ir
Leila
Nouri
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
leila_nouri@math.usb.ac.ir
10.29252/cgasa.12.1.175
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 $(PWP)$. <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 $(PWP_{ssc})$.
$S$-act,Flatness properties,Condition $(PWP_{ssc})$,semi-cancellative,$e$-cancellative
https://cgasa.sbu.ac.ir/article_85729.html
https://cgasa.sbu.ac.ir/article_85729_9d3888f3fd18b864c1967d267a21ae2c.pdf