Cover for Vol. 11, No. 1.
text
article
2019
eng
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
https://cgasa.sbu.ac.ir/article_87123_9b4005a028f1d0cdf8e4f5bdbb19c893.pdf
Celebrating Professor George A. Grätzer
Gábor
Czédli
University of Szeged, Bolyai Institute, Szeged, Aradi vértanúk tere 1, Hungary 6720.
author
text
article
2019
eng
It is a great honor for me to write a few introductory words to the present volume of CGASA dedicated to Professor George A. Grätzer. 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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
1
10
https://cgasa.sbu.ac.ir/article_87121_06b0f9dca7522e43041de4323fcf6938.pdf
An interview with George A. Grätzer
Gábor
Czédli
University of Szeged, Bolyai Institute, Szeged, Aradi vértanúk tere 1, Hungary 6720.
author
text
article
2019
eng
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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
11
18
https://cgasa.sbu.ac.ir/article_87120_1fd822e28d8bb91a38c64df9fcb0b807.pdf
The function ring functors of pointfree topology revisited
Bernhard
Banaschewski
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.
author
text
article
2019
eng
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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
19
32
https://cgasa.sbu.ac.ir/article_87117_d2e1481a97bb9235d4ea8ec563e32744.pdf
dx.doi.org/10.29252/cgasa.11.1.19
On semi weak factorization structures
Azadeh
Ilaghi-Hosseini
Department of Pure Mathematics, Faculty of Math and Computer, Shahid Bahonar University of Kerman
author
Seyed Shahin
Mousavi Mirkalai
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
author
Naser
Hosseini
Department of Pure Mathematics, Faculty of Math and Computers, Shahid Bahonar University of Kerman, Kerman, Iran
author
text
article
2019
eng
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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
33
56
https://cgasa.sbu.ac.ir/article_76603_4b608595da687086c978149f3a596b28.pdf
dx.doi.org/10.29252/cgasa.11.1.33
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
Gábor
Czédli
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary
author
Árpád
Kurusa
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720
author
text
article
2019
eng
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 $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $\,\mathcal U_{1-k}$ is included in the convex hull of $\,\mathcal U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles.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. 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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
57
92
https://cgasa.sbu.ac.ir/article_82639_995ede57b706f33c6488407d8fdd492d.pdf
dx.doi.org/10.29252/cgasa.11.1.57
The categories of lattice-valued maps, equalities, free objects, and $\mathcal C$-reticulation
Abolghasem
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
author
text
article
2019
eng
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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
93
112
https://cgasa.sbu.ac.ir/article_87118_fdc1919782d40300997d11b44b33109b.pdf
dx.doi.org/10.29252/cgasa.11.1.93
Another proof of Banaschewski's surjection theorem
Dharmanand
Baboolal
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
author
Jorge
Picado
Department of Mathematics
University of Coimbra
PORTUGAL
author
Ales
Pultr
Department of Applied Mathematics and ITI, MFF, Charles University,
Malostranske nam. 24, 11800 Praha 1, Czech Republic
author
text
article
2019
eng
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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
113
130
https://cgasa.sbu.ac.ir/article_76726_f134d7becf0d86ec81e2ee5972440080.pdf
dx.doi.org/10.29252/cgasa.11.1.113
Intersection graphs associated with semigroup acts
Abdolhossein
Delfan
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran,
author
Hamid
Rasouli
Department of Mathematics, Science and Research Branch, Islamic
Azad University, Tehran, Iran
author
Abolfazl
Tehranian
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
author
text
article
2019
eng
< 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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
131
148
https://cgasa.sbu.ac.ir/article_76602_f65aa5a84b61acf36853ad0f3af7d2f7.pdf
dx.doi.org/10.29252/cgasa.11.1.131
Completeness results for metrized rings and lattices
George
Bergman
University of California, Berkeley
author
text
article
2019
eng
The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, $\{0\})$ that are closed under the natural metric, but has no prime 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. 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 either the inequality $d(x\vee y,\,x\vee z)\leq d(y,z)$ or the inequality $d(x\wedge y,x\wedge z)\leq d(y,z),$ 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. We show by example that if the above inequalities are replaced by the weaker conditions $d(x,\,x\vee y)\leq d(x,y),$ respectively $d(x,\,x\wedge y)\leq d(x,y),$ the completeness conclusion can fail. We end with two open questions.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
149
168
https://cgasa.sbu.ac.ir/article_82638_41c589d665953b3ab2260903c95697c4.pdf
dx.doi.org/10.29252/cgasa.11.1.149
(r,t)-injectivity in the category $S$-Act
Mahdieh
Haddadi
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
author
Seyed Mojtaba
Naser Sheykholislami
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
author
text
article
2019
eng
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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
169
196
https://cgasa.sbu.ac.ir/article_76601_35b108e0967457882abe5232f68aa727.pdf
dx.doi.org/10.29252/cgasa.11.1.169
Frankl's Conjecture for a subclass of semimodular lattices
Vinayak
Joshi
Department of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune - 411007
author
Baloo
Waphare
Department of Mathematics, Savitribai Phule Pune University,
Pune-411007, India.
author
text
article
2019
eng
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.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
197
206
https://cgasa.sbu.ac.ir/article_85730_335445da865e1a5e147830cee5b78a6e.pdf
dx.doi.org/10.29252/cgasa.11.1.197
Persian Abstracts, Vol. 11, No. 1.
text
article
2019
eng
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
11
v.
Special Issue Dedicated to Prof. George A. Grätzer
no.
2019
207
222
https://cgasa.sbu.ac.ir/article_87122_e86c8ddae16fa161105c8e5e80620091.pdf