Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
The function ring functors of pointfree topology revisited
19
32
EN
Bernhard
Banaschewski
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.
iscoe@math.mcmaster.ca
10.29252/cgasa.11.1.19
This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued functions, and for the related functors ${mathfrak R}^*$ and ${mathfrak Z}^*$ corresponding to the bounded functions. Further it is shown that some familiar facts concerning these functors are simple consequences of the present results.
Completely regular frames,zero dimensional frames,completely regular $sigma$-frames,zero dimensional $sigma$-frames,real-valued continuous functions and integer-valued continuous functions on frames
https://cgasa.sbu.ac.ir/article_87117.html
https://cgasa.sbu.ac.ir/article_87117_d2e1481a97bb9235d4ea8ec563e32744.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
On semi weak factorization structures
33
56
EN
Azadeh
Ilaghi-Hosseini
Department of Pure Mathematics, Faculty of Math and Computer, Shahid Bahonar University of Kerman
a.ilaghi@math.uk.ac.ir
Seyed Shahin
Mousavi Mirkalai
0000-0002-2904-7692
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
smousavi@uk.ac.ir
Naser
Hosseini
Department of Pure Mathematics, Faculty of Math and Computers, Shahid Bahonar University of Kerman, Kerman, Iran
nhoseini@uk.ac.ir
10.29252/cgasa.11.1.33
In this article the notions of semi weak orthogonality and semi weak factorization structure in a category $mathcal X$ are introduced. Then the relationship between semi weak factorization structures and quasi right (left) and weak factorization structures is given. The main result is a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structures by natural isomorphisms.
Quasi right (left) factorization structure,(semi weak) orthogonality,(semi weak),factorization structure
https://cgasa.sbu.ac.ir/article_76603.html
https://cgasa.sbu.ac.ir/article_76603_4b608595da687086c978149f3a596b28.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
57
92
EN
Gábor
Czédli
0000-0001-9990-3573
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary
czedli@math.u-szeged.hu
Árpád
Kurusa
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720
kurusa@math.u-szeged.hu
10.29252/cgasa.11.1.57
K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.<br />Here we prove the existence of such a $j$ and $k$ for the more general case where $,mathcal U_0$ and $,mathcal U_1$ are compact sets in the plane such that $,mathcal U_1$ is obtained from $,mathcal U_0$ by a positive homothety or by a translation. <br />Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Grätzer and E. Knapp, lead to our result.
Congruence lattice,planar semimodular lattice,convex hull,compact set,linebreak circle,combinatorial geometry,abstract convex geometry,anti-exchange property
https://cgasa.sbu.ac.ir/article_82639.html
https://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
The categories of lattice-valued maps, equalities, free objects, and $mathcal C$-reticulation
93
112
EN
Abolghasem
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
akarimi@gorganiau.ac.ir
10.29252/cgasa.11.1.93
In this paper, we study the concept of $mathcal C$-reticulation for the category $mathcal C$ whose objects are lattice-valued maps. The relation between the free objects in $mathcal C$ and the $mathcal C$-reticulation of rings and modules is discussed. Also, a method to construct $mathcal C$-reticulation is presented, in the case where $mathcal C$ is equational. Some relations between the concepts reticulation and satisfying equalities and inequalities are studied.
Free object,$ell$-ring,$ell$-module,frame,cozero map,semi-cozero map,the $F$-Zariski topology,$mathcal C$-reticulation,lattice-valued map
https://cgasa.sbu.ac.ir/article_87118.html
https://cgasa.sbu.ac.ir/article_87118_fdc1919782d40300997d11b44b33109b.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
Another proof of Banaschewski's surjection theorem
113
130
EN
Dharmanand
Baboolal
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
baboolald@ukzn.ac.za
Jorge
Picado
Department of Mathematics
University of Coimbra
PORTUGAL
picado@mat.uc.pt
Ales
Pultr
Department of Applied Mathematics and ITI, MFF, Charles University,
Malostranske nam. 24, 11800 Praha 1, Czech Republic
pultr@kam.mff.cuni.cz
10.29252/cgasa.11.1.113
We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform sublocale can be extended to a (regular) Cauchy point on the larger (quasi-)uniform frame.
Frame (locale),sublocale,uniform frame,quasi-uniform frame,uniform embedding,complete uniform frame,completion,Cauchy map,Cauchy filter,Cauchy complete
https://cgasa.sbu.ac.ir/article_76726.html
https://cgasa.sbu.ac.ir/article_76726_f134d7becf0d86ec81e2ee5972440080.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
Intersection graphs associated with semigroup acts
131
148
EN
Abdolhossein
Delfan
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran,
a.delfan@khoiau.ac.ir
Hamid
Rasouli
Department of Mathematics, Science and Research Branch, Islamic
Azad University, Tehran, Iran
hrasouli@srbiau.ac.ir
Abolfazl
Tehranian
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
tehranian@srbiau.ac.ir
10.29252/cgasa.11.1.131
< p>The intersection graph $\mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $\mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the finiteness of each of the clique number, the chromatic number, and the degree of some or all vertices in $\mathbb{Int}(A)$ is equivalent to the finiteness of the number of subacts of $A$. Finally, we determine the clique number of the graphs of certain classes of $S$-acts.
$S$-act,intersection graph,Chromatic number,Clique number,weakly perfect graph
https://cgasa.sbu.ac.ir/article_76602.html
https://cgasa.sbu.ac.ir/article_76602_f65aa5a84b61acf36853ad0f3af7d2f7.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
Completeness results for metrized rings and lattices
149
168
EN
George
M.
Bergman
0000-0003-4027-7293
University of California, Berkeley
gbergman@math.berkeley.edu
10.29252/cgasa.11.1.149
The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper <em>radical</em> ideals (for example, ${0})$ that are closed under the natural metric, but has no <em>prime</em> ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Together, these facts answer a question posed by J. Gleason. From this example, rings of arbitrary characteristic with the same properties are obtained. <br />The result that $B$ is complete in its metric is generalized to show that if $L$ is a lattice given with a metric satisfying identically <em>either</em> the inequality $d(xvee y,,xvee z)leq d(y,z)$ <em>or</em> the inequality $d(xwedge y,xwedge z)leq d(y,z),$ and if in $L$ every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in $L$ converges; that is, $L$ is complete as a metric space. <br />We show by example that if the above inequalities are replaced by the weaker conditions $d(x,,xvee y)leq d(x,y),$ respectively $d(x,,xwedge y)leq d(x,y),$ the completeness conclusion can fail. <br />We end with two open questions.
Complete topological ring without closed prime ideals,measurable sets modulo sets of measure zero,lattice complete under a metric
https://cgasa.sbu.ac.ir/article_82638.html
https://cgasa.sbu.ac.ir/article_82638_41c589d665953b3ab2260903c95697c4.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
(r,t)-injectivity in the category $S$-Act
169
196
EN
Mahdieh
Haddadi
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
m.haddadi@semnan.ac.ir
Seyed Mojtaba
Naser Sheykholislami
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
s.m.naser@semnan.ac.ir
10.29252/cgasa.11.1.169
In this paper, we show that injectivity with respect to the class $mathcal{D}$ of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category well-behaves. Indeed, if $mathcal{M}$ is a subclass of monomorphisms, $mathcal{M}cap mathcal{D}$-injectivity well-behaves. We also introduce the notion of $(r,t)$-injectivity in the category {bf S-Act}, where $r$ and $t$ are Hoehnke radicals, and discuss whether this kind of injectivity well-behaves.
Injectivity,$S$-act,Hoehnke radical
https://cgasa.sbu.ac.ir/article_76601.html
https://cgasa.sbu.ac.ir/article_76601_35b108e0967457882abe5232f68aa727.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
11
Special Issue Dedicated to Prof. George A. Grätzer
2019
07
01
Frankl's Conjecture for a subclass of semimodular lattices
197
206
EN
Vinayak
Joshi
0000-0001-9105-4634
Department of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune - 411007
vinayakjoshi111@yahoo.com
Baloo
Waphare
Department of Mathematics, Savitribai Phule Pune University,
Pune-411007, India.
waphare@yahoo.com
10.29252/cgasa.11.1.197
In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices having breadth at most two. We provide a very short proof of the Conjecture for the class of lattices having breadth at most two. This generalizes the results of Joshi, Waphare and Kavishwar as well as Czédli and Schmidt.
Union-Closed Sets Conjecture,Frankl's Conjecture,semimodular lattice,adjunct operation
https://cgasa.sbu.ac.ir/article_85730.html
https://cgasa.sbu.ac.ir/article_85730_335445da865e1a5e147830cee5b78a6e.pdf