eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2018-07-01
9
1
1
13
61475
Representation of $H$-closed monoreflections in archimedean $ell$-groups with weak unit
Bernhard Banaschewski
1
Anthony W. Hager
ahager@wesleyan.edu
2
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada.
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
The category of the title is called $mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $mathcal{A}$, $Hmathcal{A}$ consists of all homomorphic images of $mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(mathcal{R}, r)$ (meaning $Hmathcal{R} = mathcal{R}$), 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 $C(T)$.
http://cgasa.sbu.ac.ir/article_61475_f777dd362fb1959c3a9aa5115a63f9a9.pdf
Archimedean $ell$-group
$H$-closed monoreflection
Yosida representation
countable composition
epicomplete
Baire functions
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2018-07-01
9
1
15
27
50749
Total graph of a $0$-distributive lattice
Shahabaddin Ebrahimi Atani
ebrahimi@guilan.ac.ir
1
Saboura Dolati Pishhesari
saboura_dolati@yahoo.com
2
Mehdi Khoramdel
mehdikhoramdel@gmail.com
3
Maryam Sedghi
ms.maryamsedghi55@gmail.com
4
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
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.
http://cgasa.sbu.ac.ir/article_50749_c43feee35e55c325b3f13fa98313523d.pdf
Lattice
minimal prime ideal
zero-divisor graph
total graph
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2018-07-01
9
1
29
58
50747
On lifting of biadjoints and lax algebras
Fernando Lucatelli Nunes
lucatellinunes@student.uc.pt
1
CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal.
Given a pseudomonad $mathcal{T} $ on a $2$-category $mathfrak{B} $, if a right biadjoint $mathfrak{A}tomathfrak{B} $ has a lifting to the pseudoalgebras $mathfrak{A}tomathsf{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 <em>biadjoint triangle theorem</em> 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} tomathsf{Lax}textrm{-}mathcal{T}textrm{-}mathsf{Alg} _ell $ the inclusion, if $R: mathfrak{A}tomathfrak{B} $ is right biadjoint and has a lifting $J: mathfrak{A}to mathsf{Lax}textrm{-}mathcal{T}textrm{-}mathsf{Alg} $, then $ellcirc J$ is right biadjoint as well provided that $mathfrak{A} $ has some needed weighted bicolimits. In order to prove such result, we study <em>descent objects</em> and <em>lax descent objects</em>. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence.
http://cgasa.sbu.ac.ir/article_50747_e7751692a69d525e49259ebe2763142f.pdf
Lax algebras
pseudomonads
biadjunctions
adjoint triangles
lax descent objects
descent categories
weighted bi(co)limits
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2018-07-01
9
1
59
75
50745
Pointfree topology version of image of real-valued continuous functions
Abolghasem Karimi Feizabadi
bolghasem.karimi.f@gmail.com
1
Ali Akbar Estaji
aa_estaji@yahoo.com
2
Maryam Robat Sarpoushi
m.sarpooshi@yahoo.com
3
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.
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 $C_c(X).$<br />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 <em>overlap</em>. 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 $f:Xrightarrowmathbb R$ and $ S subseteq mathbb R$.
http://cgasa.sbu.ac.ir/article_50745_d90d55e08316779860740922b0388294.pdf
frame
ring of real-valued continuous functions
countable image
$f$-ring
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2018-07-01
9
1
77
138
58262
Convergence and quantale-enriched categories
Dirk Hofmann
dirk@ua.pt
1
Carla D. Reis
carla.reis@estgoh.ipc.pt
2
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.
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.
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.
http://cgasa.sbu.ac.ir/article_58262_bab8553989f148c1daf7939ffd5b9f4d.pdf
Ordered compact Hausdorff space
metric space
approach space
sober space
Cauchy completness
quantale-enriched category
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2018-07-01
9
1
139
161
50748
Convex $L$-lattice subgroups in $L$-ordered groups
Rajabali Borzooei
borzooei@sbu.ac.ir
1
Fateme Hosseini
hoseini_nm@yahoo.com
2
Omid Zahiri
zahiri@protonmail.com
3
Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.
Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.
University of Applied Science and Technology, Tehran, Iran
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.
http://cgasa.sbu.ac.ir/article_50748_0ee3783313053dea8791d1990de4c8e2.pdf
$L$-ordered group
convex $L$-subgroup
(normal) convex $L$-lattice subgroup