Cover for Vol. 9, No. 1.
text
article
2018
eng
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
https://cgasa.sbu.ac.ir/article_65930_1e6b40c8fcf8398bfb33cedee74afaa6.pdf
Representation of $H$-closed monoreflections in archimedean $\ell$-groups with weak unit
Bernhard
Banaschewski
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada.
author
Anthony W.
Hager
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
author
text
article
2018
eng
The category of the title is called $\mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $\mathcal{A}$, $H\mathcal{A}$ consists of all homomorphic images of $\mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(\mathcal{R}, r)$ (meaning $H\mathcal{R} = \mathcal{R}$), about which we show ({\em inter alia}): $A \in \mathcal{A}$ if and only if $A$ is a countably up-directed union from $H\{rF(\omega)\}$. The meaning of this is then analyzed for two important cases: the maximum essential monoreflection $r = c^{3}$, where $c^{3}F(\omega) = C(\RR^{\omega})$, and $C \in H\{c(\RR^{\omega})\}$ means $C = C(T)$, for $T$ a closed subspace of $\RR^{\omega}$; the epicomplete, and maximum, monoreflection, $r = \beta$, where $\beta F(\omega) = B(\RR^{\omega})$, the Baire functions, and $E \in H\{B(\RR^{\omega})\}$ means $E$ is {\em an} epicompletion (not ``the'') of such a $C(T)$.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
1
13
https://cgasa.sbu.ac.ir/article_61475_f777dd362fb1959c3a9aa5115a63f9a9.pdf
dx.doi.org/10.29252/cgasa.9.1.1
Total graph of a $0$-distributive lattice
Shahabaddin
Ebrahimi Atani
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
author
Saboura
Dolati Pishhesari
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
author
Mehdi
Khoramdel
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
author
Maryam
Sedghi
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
author
text
article
2018
eng
Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${\rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${\rm T}(G (£))$. It is the graph with all elements of £ as vertices, and for distinct $x, y \in £$, the vertices $x$ and $y$ are adjacent if and only if $x \vee y \in {\rm Z}(£)$. The basic properties of the graph ${\rm T}(G (£))$ and its subgraphs are studied. We investigate the properties of the total graph of $0$-distributive lattices as diameter, girth, clique number, radius, and the independence number.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
15
27
https://cgasa.sbu.ac.ir/article_50749_c43feee35e55c325b3f13fa98313523d.pdf
dx.doi.org/10.29252/cgasa.9.1.15
On lifting of biadjoints and lax algebras
Fernando
Lucatelli Nunes
CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal.
author
text
article
2018
eng
Given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then this lifting is also right biadjoint provided that $\mathfrak{A} $ has codescent objects. In this paper, we give general results on lifting of biadjoints. As a consequence, we get a biadjoint triangle theorem which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. In the context of lax algebras, denoting by $\ell :\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} \to\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} _\ell $ the inclusion, if $R: \mathfrak{A}\to\mathfrak{B} $ is right biadjoint and has a lifting $J: \mathfrak{A}\to \mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $, then $\ell\circ J$ is right biadjoint as well provided that $\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such result, we study descent objects and lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
29
58
https://cgasa.sbu.ac.ir/article_50747_e7751692a69d525e49259ebe2763142f.pdf
dx.doi.org/10.29252/cgasa.9.1.29
Pointfree topology version of image of real-valued continuous functions
Abolghasem
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
author
Ali Akbar
Estaji
Faculty of Mathematics and Computer Sciences,
Hakim Sabzevari University, Sabzevar, Iran.
author
Maryam
Robat Sarpoushi
Faculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.
author
text
article
2018
eng
Let $ { \mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree version of $C_c(X).$The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in $ {\mathcal{R}} L$. In particular, we will introduce the pointfree version of the ring $C_c(X)$. We define a relation from $ {\mathcal{R}} L$ into the power set of $\mathbb R$, namely overlap . Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as $\mathop{\hbox{Im}} (f) \subseteq S$ for every continuous function $f:X\rightarrow\mathbb R$ and $ S \subseteq \mathbb R$.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
59
75
https://cgasa.sbu.ac.ir/article_50745_d90d55e08316779860740922b0388294.pdf
dx.doi.org/10.29252/cgasa.9.1.59
Convergence and quantale-enriched categories
Dirk
Hofmann
Center for Research and Development in Mathematics and Applications,
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.
author
Carla
D. Reis
Polytechnic Institute of Coimbra, College of Management and Technology
of Oliveira do Hospital, 3400-124 Oliveira do Hospital, Portugal; and Center
for Research and Development in Mathematics and Applications, University of
Aveiro, Portugal.
author
text
article
2018
eng
Generalising Nachbin's theory of ``topology and order'', in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\V$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
77
138
https://cgasa.sbu.ac.ir/article_58262_bab8553989f148c1daf7939ffd5b9f4d.pdf
dx.doi.org/10.29252/cgasa.9.1.77
Convex $L$-lattice subgroups in $L$-ordered groups
Rajabali
Borzooei
Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.
author
Fateme
Hosseini
Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.
author
Omid
Zahiri
University of Applied Science and Technology, Tehran, Iran
author
text
article
2018
eng
In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result about the set of all convex subgroups of a lattice ordered group and prove that $C(G)$, the set of all convex $L$-lattice subgroups of an $L$-ordered group $G$, is an $L$-complete lattice on height one. Then we use these objects to construct the quotient $L$-ordered groups and state some related results.
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
139
161
https://cgasa.sbu.ac.ir/article_50748_0ee3783313053dea8791d1990de4c8e2.pdf
dx.doi.org/10.29252/cgasa.9.1.139
Persian Abstracts, Vol. 9, No. 1.
text
article
2018
eng
Categories and General Algebraic Structures with Applications
Shahid Beheshti University
2345-5853
9
v.
1
no.
2018
172
182
https://cgasa.sbu.ac.ir/article_65931_d0ea46be4127864d66630e8a8548674d.pdf