Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
4
1
2016
02
01
Birkhoff's Theorem from a geometric perspective: A simple example
1
8
EN
F. William
Lawvere
Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
wlawvere@buffalo.edu
From Hilbert's theorem of zeroes, and from Noether's ideal theory, Birkhoff derived certain algebraic concepts (as explained by Tholen) that have a dual significance in general toposes, similar to their role in the original examples of algebraic geometry. I will describe a simple example that illustrates some of the aspects of this relationship. The dualization from algebra to geometry in the basic Grothendieck spirit can be accomplished (without intervention of topological spaces) by the following method, known as Isbell conjugacy.
Grothendieck spectrum,Cantor,Boole,Hilbert,Birkhoff: Existence and Sufficiency of generalized points,Reflexive Graphs
https://cgasa.sbu.ac.ir/article_12425.html
https://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
4
1
2016
02
01
Steps toward the weak higher category of weak higher categories in the globular setting
9
42
EN
Camell
Kachour
Department of Mathematics, Macquarie University, North Ryde, NSW 2109, Australia.
camell.kachour@gmail.com
We start this article by rebuilding higher operads of weak higher transformations, and correct those in cite{Cambat}. As in cite{Cambat} we propose an operadic approach for weak higher $n$-transformations, for each $ninmathbb{N}$, where such weak higher $n$-transformations are seen as algebras for specific contractible higher operads. The last chapter of this article asserts that, up to precise hypotheses, the higher operad $B^{0}_{C}$ of Batanin and the terminal higher operad $B^{0}_{S_{u}}$, both have the fractal property. In other words we isolate the precise technical difficulties behind a major problem in globular higher category theory, namely, that of proving the existence of the globular weak higher category of globular weak higher categories.
globular sets,weak higher categories,weak higher transformations,higher operads
https://cgasa.sbu.ac.ir/article_11180.html
https://cgasa.sbu.ac.ir/article_11180_b13cacfd9afe5780932141c269d0add6.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
4
1
2016
02
01
Basic notions and properties of ordered semihyperrings
43
62
EN
B.
Davvaz
Department of
Mathematics, Yazd University, Yazd, Iran.
davvaz@yazd.ac.ir
S.
Omidi
Department of
Mathematics, Yazd University, Yazd, Iran.
omidi.saber@yahoo.com
In this paper, we introduce the concept of semihyperring $(R,+,cdot)$ together with a suitable partial order $le$. Moreover, we introduce and study hyperideals in ordered semihyperrings. Simple ordered semihyperrings are defined and its characterizations are obtained. Finally, we study some properties of quasi-simple and $B$-simple ordered semihyperrings.
ordered semihyperring,hyperideal,simple,quasi-simple,$B$-simple
https://cgasa.sbu.ac.ir/article_11181.html
https://cgasa.sbu.ac.ir/article_11181_a73c9c7bbdb038f75ed62901bc042c2a.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
4
1
2016
02
01
A characterization of finitely generated multiplication modules
63
74
EN
Somayeh
Karimzadeh
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
karimzadeh@vru.ac.ir
Somayeh
Hadjirezaei
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
s.hajirezaei@vru.ac.ir
Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals of modules which are the direct sum of finitely generated module and finitely generated multiplication module are shown. Finally, we study some properties of modules that are the direct sum of multiplication modules in terms of Fitting ideals.
Fitting ideals,multiplication module,projective module
https://cgasa.sbu.ac.ir/article_12667.html
https://cgasa.sbu.ac.ir/article_12667_7069a62adca415a4a7178c2d5b4804a7.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
4
1
2016
02
01
The ring of real-continuous functions on a topoframe
75
94
EN
Ali Akbar
Estaji
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
aaestaji@hsu.ac.ir
Abolghasem
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
akarimi@gorganiau.ac.ir
Mohammad
Zarghani
Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
zarghanim@yahoo.com
A topoframe, denoted by $L_{ tau}$, is a pair $(L, tau)$ consisting of a frame $L$ and a subframe $ tau $ all of whose elements are complementary elements in $L$. In this paper, we define and study the notions of a $tau $-real-continuous function on a frame $L$ and the set of real continuous functions $mathcal{R}L_tau $ as an $f$-ring. We show that $mathcal{R}L_{ tau}$ is actually a generalization of the ring $C(X)$ of all real-valued continuous functions on a completely regular Hausdorff space $X$. In addition, we show that $mathcal{R}L_{ tau}$ is isomorphic to a sub-$f$-ring of $mathcal{R}tau .$ Let ${tau}$ be a topoframe on a frame $L$. The frame map $alphainmathcal{R}tau $ is called $L$-{it extendable} real continuous function if and only if for every $rin mathbb{R}$, $bigvee^{L}_{rin mathbb R} (alpha(-,r)veealpha(r,-))'=top.$ Finally, we prove that $mathcal{R}^{L}{tau}cong mathcal{R}L_{tau}$ as $f$-rings, where $mathcal{R}^{L}{tau}$ is the set all of $L$-extendable real continuous functions of $ mathcal{R}tau $.
frame,Topoframe,Ring of real continuous
functions,Archimedean ring,$f$-ring
https://cgasa.sbu.ac.ir/article_13184.html
https://cgasa.sbu.ac.ir/article_13184_2f80da4a155068ca432d536a7217a6ab.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
4
1
2016
02
01
On zero divisor graph of unique product monoid rings over Noetherian reversible ring
95
114
EN
Ebrahim
Hashemi
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
eb_hashemi@shahroodut.ac.i
Abdollah
Alhevaz
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
a.alhevaz@shahroodut.ac.ir
Eshag
Yoonesian
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
eshagh.eshag@gmail.com
Let $R$ be an associative ring with identity and $Z^*(R)$ be its set of non-zero zero divisors. The zero-divisor graph of $R$, denoted by $Gamma(R)$, is the graph whose vertices are the non-zero zero-divisors of $R$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$. In this paper, we bring some results about undirected zero-divisor graph of a monoid ring over reversible right (or left) Noetherian ring $R$. We essentially classify the diameter-structure of this graph and show that $0leq mbox{diam}(Gamma(R))leq mbox{diam}(Gamma(R[M]))leq 3$. Moreover, we give a characterization for the possible diam$(Gamma(R))$ and diam$(Gamma(R[M]))$, when $R$ is a reversible Noetherian ring and $M$ is a u.p.-monoid. Also, we study relations between the girth of $Gamma(R)$ and that of $Gamma(R[M])$.
Zero-divisor graphs,diameter,Girth,Reversible rings,Polynomial rings,Unique product monoids,Monoid rings
https://cgasa.sbu.ac.ir/article_13185.html
https://cgasa.sbu.ac.ir/article_13185_9afc1a95b9340cdc8d14a1cee3b2fe5c.pdf