eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
19
32
87117
The function ring functors of pointfree topology revisited
Bernhard Banaschewski
iscoe@math.mcmaster.ca
1
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.
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.
http://cgasa.sbu.ac.ir/article_87117_d2e1481a97bb9235d4ea8ec563e32744.pdf
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
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
33
56
76603
On semi weak factorization structures
Azadeh Ilaghi-Hosseini
a.ilaghi@math.uk.ac.ir
1
Seyed Shahin Mousavi Mirkalai
smousavi@uk.ac.ir
2
Naser Hosseini
nhoseini@uk.ac.ir
3
Department of Pure Mathematics, Faculty of Math and Computer, Shahid Bahonar University of Kerman
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of Math and Computers, Shahid Bahonar University of Kerman, Kerman, Iran
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.
http://cgasa.sbu.ac.ir/article_76603_4b608595da687086c978149f3a596b28.pdf
Quasi right (left) factorization structure
(semi weak) orthogonality
(semi weak)
factorization structure
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
57
92
82639
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
Gábor Czédli
czedli@math.u-szeged.hu
1
Árpád Kurusa
kurusa@math.u-szeged.hu
2
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720
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.
http://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf
Congruence lattice
planar semimodular lattice
convex hull
compact set
linebreak circle
combinatorial geometry
abstract convex geometry
anti-exchange property
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
93
112
87118
The categories of lattice-valued maps, equalities, free objects, and $mathcal C$-reticulation
Abolghasem Karimi Feizabadi
akarimi@gorganiau.ac.ir
1
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
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.
http://cgasa.sbu.ac.ir/article_87118_fdc1919782d40300997d11b44b33109b.pdf
Free object
$ell$-ring
$ell$-module
frame
cozero map
semi-cozero map
the $F$-Zariski topology
$mathcal C$-reticulation
lattice-valued map
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
113
130
76726
Another proof of Banaschewski's surjection theorem
Dharmanand Baboolal
baboolald@ukzn.ac.za
1
Jorge Picado
picado@mat.uc.pt
2
Ales Pultr
pultr@kam.mff.cuni.cz
3
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
Department of Mathematics University of Coimbra PORTUGAL
Department of Applied Mathematics and ITI, MFF, Charles University, Malostranske nam. 24, 11800 Praha 1, Czech Republic
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.
http://cgasa.sbu.ac.ir/article_76726_f134d7becf0d86ec81e2ee5972440080.pdf
Frame (locale)
sublocale
uniform frame
quasi-uniform frame
uniform embedding
complete uniform frame
completion
Cauchy map
Cauchy filter
Cauchy complete
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
131
148
76602
Intersection graphs associated with semigroup acts
Abdolhossein Delfan
a.delfan@khoiau.ac.ir
1
Hamid Rasouli
hrasouli@srbiau.ac.ir
2
Abolfazl Tehranian
tehranian@srbiau.ac.ir
3
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran,
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
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.
http://cgasa.sbu.ac.ir/article_76602_f65aa5a84b61acf36853ad0f3af7d2f7.pdf
$S$-act
intersection graph
Chromatic number
Clique number
weakly perfect graph
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
149
168
82638
Completeness results for metrized rings and lattices
George Bergman
gbergman@math.berkeley.edu
1
University of California, Berkeley
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.
http://cgasa.sbu.ac.ir/article_82638_41c589d665953b3ab2260903c95697c4.pdf
Complete topological ring without closed prime ideals
measurable sets modulo sets of measure zero
lattice complete under a metric
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
169
196
76601
(r,t)-injectivity in the category $S$-Act
Mahdieh Haddadi
m.haddadi@semnan.ac.ir
1
Seyed Mojtaba Naser Sheykholislami
s.m.naser@semnan.ac.ir
2
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
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.
http://cgasa.sbu.ac.ir/article_76601_35b108e0967457882abe5232f68aa727.pdf
Injectivity
$S$-act
Hoehnke radical
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2019-07-01
11
Special Issue Dedicated to Prof. George A. Grätzer
197
206
85730
Frankl's Conjecture for a subclass of semimodular lattices
Vinayak Joshi
vinayakjoshi111@yahoo.com
1
Baloo Waphare
waphare@yahoo.com
2
Department of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune - 411007
Department of Mathematics, Savitribai Phule Pune University, Pune-411007, India.
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.
http://cgasa.sbu.ac.ir/article_85730_335445da865e1a5e147830cee5b78a6e.pdf
Union-Closed Sets Conjecture
Frankl's Conjecture
semimodular lattice
adjunct operation