2019-10-21T04:38:47Z
http://cgasa.sbu.ac.ir/?_action=export&rf=summon&issue=11595
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
Cover for Vol. 11, No. 1.
2019
07
01
http://cgasa.sbu.ac.ir/article_87123_9b4005a028f1d0cdf8e4f5bdbb19c893.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
Celebrating Professor George A. Grätzer
Gábor
Czédli
It is a great honor for me to write a few introductory words to the present volume of CGASA <strong>dedicated to Professor George A. Grätzer</strong>. The occasion for this dedication is that we are celebrating two anniversaries in 2018 related to him. Namely,
(A1) it was 55 years ago that the Grätzer--Schmidt Theorem was published, and
(A2) it was 40 years ago that G. Grätzer's General Lattice Theory, which immediately became the Book in lattice theory for decades, appeared.
George Grätzer
lattice theory
Grätzer--Schmidt Theorem
2019
07
01
1
10
http://cgasa.sbu.ac.ir/article_87121_06b0f9dca7522e43041de4323fcf6938.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
An interview with George A. Grätzer
Gábor
Czédli
This interview was conducted in the second half of May, 2018. Both George Grätzer and the author were at home, in Toronto and Szeged, respectively. They communicated via a lot of e-mails and a few phone calls.
George A. Grätzer
Interview
2019
07
01
11
18
http://cgasa.sbu.ac.ir/article_87120_1fd822e28d8bb91a38c64df9fcb0b807.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
The function ring functors of pointfree topology revisited
Bernhard
Banaschewski
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
2019
07
01
19
32
http://cgasa.sbu.ac.ir/article_87117_d2e1481a97bb9235d4ea8ec563e32744.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
On semi weak factorization structures
Azadeh
Ilaghi-Hosseini
Seyed Shahin
Mousavi Mirkalai
Naser
Hosseini
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
2019
07
01
33
56
http://cgasa.sbu.ac.ir/article_76603_4b608595da687086c978149f3a596b28.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
Gábor
Czédli
Árpád
Kurusa
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
2019
07
01
57
92
http://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
The categories of lattice-valued maps, equalities, free objects, and $mathcal C$-reticulation
Abolghasem
Karimi Feizabadi
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
2019
07
01
93
112
http://cgasa.sbu.ac.ir/article_87118_fdc1919782d40300997d11b44b33109b.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
Another proof of Banaschewski's surjection theorem
Dharmanand
Baboolal
Jorge
Picado
Ales
Pultr
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
2019
07
01
113
130
http://cgasa.sbu.ac.ir/article_76726_f134d7becf0d86ec81e2ee5972440080.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
Intersection graphs associated with semigroup acts
Abdolhossein
Delfan
Hamid
Rasouli
Abolfazl
Tehranian
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
2019
07
01
131
148
http://cgasa.sbu.ac.ir/article_76602_f65aa5a84b61acf36853ad0f3af7d2f7.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
Completeness results for metrized rings and lattices
George
Bergman
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
2019
07
01
149
168
http://cgasa.sbu.ac.ir/article_82638_41c589d665953b3ab2260903c95697c4.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
(r,t)-injectivity in the category $S$-Act
Mahdieh
Haddadi
Seyed Mojtaba
Naser Sheykholislami
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
2019
07
01
169
196
http://cgasa.sbu.ac.ir/article_76601_35b108e0967457882abe5232f68aa727.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
Frankl's Conjecture for a subclass of semimodular lattices
Vinayak
Joshi
Baloo
Waphare
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
2019
07
01
197
206
http://cgasa.sbu.ac.ir/article_85730_335445da865e1a5e147830cee5b78a6e.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2019
11
Special Issue Dedicated to Prof. George A. Grätzer
Persian Abstracts, Vol. 11, No. 1.
2019
07
01
207
222
http://cgasa.sbu.ac.ir/article_87122_e86c8ddae16fa161105c8e5e80620091.pdf