eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2016-02-01
4
1
1
8
12425
Birkhoff's Theorem from a geometric perspective: A simple example
F. William Lawvere
wlawvere@buffalo.edu
1
Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
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.
http://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.pdf
Grothendieck spectrum
Cantor
Boole
Hilbert
Birkhoff: Existence and Sufficiency of generalized points
Reflexive Graphs
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2016-02-01
4
1
9
42
11180
Steps toward the weak higher category of weak higher categories in the globular setting
Camell Kachour
camell.kachour@gmail.com
1
Department of Mathematics, Macquarie University, North Ryde, NSW 2109, Australia.
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.
http://cgasa.sbu.ac.ir/article_11180_b13cacfd9afe5780932141c269d0add6.pdf
globular sets
weak higher categories
weak higher transformations
higher operads
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2016-02-01
4
1
43
62
11181
Basic notions and properties of ordered semihyperrings
B. Davvaz
davvaz@yazd.ac.ir
1
S. Omidi
omidi.saber@yahoo.com
2
Department of Mathematics, Yazd University, Yazd, Iran.
Department of Mathematics, Yazd University, Yazd, Iran.
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.
http://cgasa.sbu.ac.ir/article_11181_a73c9c7bbdb038f75ed62901bc042c2a.pdf
ordered semihyperring
hyperideal
simple
quasi-simple
$B$-simple
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2016-02-01
4
1
63
74
12667
A characterization of finitely generated multiplication modules
Somayeh Karimzadeh
karimzadeh@vru.ac.ir
1
Somayeh Hadjirezaei
s.hajirezaei@vru.ac.ir
2
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
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.
http://cgasa.sbu.ac.ir/article_12667_7069a62adca415a4a7178c2d5b4804a7.pdf
Fitting ideals
multiplication module
projective module
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2016-02-01
4
1
75
94
13184
The ring of real-continuous functions on a topoframe
Ali Akbar Estaji
aaestaji@hsu.ac.ir
1
Abolghasem Karimi Feizabadi
akarimi@gorganiau.ac.ir
2
Mohammad Zarghani
zarghanim@yahoo.com
3
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
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 $.
http://cgasa.sbu.ac.ir/article_13184_2f80da4a155068ca432d536a7217a6ab.pdf
Frame
Topoframe
Ring of real continuous
functions
Archimedean ring
$f$-ring
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2016-02-01
4
1
95
114
13185
On zero divisor graph of unique product monoid rings over Noetherian reversible ring
Ebrahim Hashemi
eb_hashemi@shahroodut.ac.i
1
Abdollah Alhevaz
a.alhevaz@shahroodut.ac.ir
2
Eshag Yoonesian
eshagh.eshag@gmail.com
3
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.
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 $Gamma(R[M])$.
http://cgasa.sbu.ac.ir/article_13185_9afc1a95b9340cdc8d14a1cee3b2fe5c.pdf
Zero-divisor graphs
diameter
Girth
Reversible rings
Polynomial rings
Unique product monoids
Monoid rings