2021-07-26T05:43:19Z
https://cgasa.sbu.ac.ir/?_action=export&rf=summon&issue=8377
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
Cover for Vol. 9, No. 1.
2018
07
01
https://cgasa.sbu.ac.ir/article_65930_1e6b40c8fcf8398bfb33cedee74afaa6.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
Representation of $H$-closed monoreflections in archimedean $ell$-groups with weak unit
Bernhard
Banaschewski
Anthony W.
Hager
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 $C(T)$.
Archimedean $ell$-group
$H$-closed monoreflection
Yosida representation
countable composition
epicomplete
Baire functions
2018
07
01
1
13
https://cgasa.sbu.ac.ir/article_61475_f777dd362fb1959c3a9aa5115a63f9a9.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
Total graph of a $0$-distributive lattice
Shahabaddin
Ebrahimi Atani
Saboura
Dolati Pishhesari
Mehdi
Khoramdel
Maryam
Sedghi
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.
Lattice
minimal prime ideal
zero-divisor graph
total graph
2018
07
01
15
27
https://cgasa.sbu.ac.ir/article_50749_c43feee35e55c325b3f13fa98313523d.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
On lifting of biadjoints and lax algebras
Fernando
Lucatelli Nunes
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.
Lax algebras
pseudomonads
biadjunctions
adjoint triangles
lax descent objects
descent categories
weighted bi(co)limits
2018
07
01
29
58
https://cgasa.sbu.ac.ir/article_50747_e7751692a69d525e49259ebe2763142f.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
Pointfree topology version of image of real-valued continuous functions
Abolghasem
Karimi Feizabadi
Ali Akbar
Estaji
Maryam
Robat Sarpoushi
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$.
frame
ring of real-valued continuous functions
countable image
$f$-ring
2018
07
01
59
75
https://cgasa.sbu.ac.ir/article_50745_d90d55e08316779860740922b0388294.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
Convergence and quantale-enriched categories
Dirk
Hofmann
Carla
D. Reis
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.
Ordered compact Hausdorff space
metric space
approach space
sober space
Cauchy completness
quantale-enriched category
2018
07
01
77
138
https://cgasa.sbu.ac.ir/article_58262_bab8553989f148c1daf7939ffd5b9f4d.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
Convex $L$-lattice subgroups in $L$-ordered groups
Rajabali
Borzooei
Fateme
Hosseini
Omid
Zahiri
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.
$L$-ordered group
convex $L$-subgroup
(normal) convex $L$-lattice subgroup
2018
07
01
139
161
https://cgasa.sbu.ac.ir/article_50748_0ee3783313053dea8791d1990de4c8e2.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2018
9
1
Persian Abstracts, Vol. 9, No. 1.
2018
07
01
172
182
https://cgasa.sbu.ac.ir/article_65931_d0ea46be4127864d66630e8a8548674d.pdf