Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
9
1
2018
07
01
Representation of $H$-closed monoreflections in archimedean $ell$-groups with weak unit
1
13
EN
Bernhard
Banaschewski
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada.
Anthony W.
Hager
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
ahager@wesleyan.edu
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 $C(T)$.
Archimedean $ell$-group,$H$-closed monoreflection,Yosida representation,countable composition,epicomplete,Baire functions
http://cgasa.sbu.ac.ir/article_61475.html
http://cgasa.sbu.ac.ir/article_61475_f777dd362fb1959c3a9aa5115a63f9a9.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
9
1
2018
07
01
Total graph of a $0$-distributive lattice
15
27
EN
Shahabaddin
Ebrahimi Atani
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
ebrahimi@guilan.ac.ir
Saboura
Dolati Pishhesari
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
saboura_dolati@yahoo.com
Mehdi
Khoramdel
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
mehdikhoramdel@gmail.com
Maryam
Sedghi
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
ms.maryamsedghi55@gmail.com
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
http://cgasa.sbu.ac.ir/article_50749.html
http://cgasa.sbu.ac.ir/article_50749_c43feee35e55c325b3f13fa98313523d.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
9
1
2018
07
01
On lifting of biadjoints and lax algebras
29
58
EN
Fernando
Lucatelli Nunes
CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal.
lucatellinunes@student.uc.pt
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.
Lax algebras,pseudomonads,biadjunctions,adjoint triangles,lax descent objects,descent categories,weighted bi(co)limits
http://cgasa.sbu.ac.ir/article_50747.html
http://cgasa.sbu.ac.ir/article_50747_e7751692a69d525e49259ebe2763142f.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
9
1
2018
07
01
Pointfree topology version of image of real-valued continuous functions
59
75
EN
Abolghasem
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
bolghasem.karimi.f@gmail.com
Ali Akbar
Estaji
Faculty of Mathematics and Computer Sciences,
Hakim Sabzevari University, Sabzevar, Iran.
aa_estaji@yahoo.com
Maryam
Robat Sarpoushi
Faculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.
m.sarpooshi@yahoo.com
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$.
frame,ring of real-valued continuous functions,countable image,$f$-ring
http://cgasa.sbu.ac.ir/article_50745.html
http://cgasa.sbu.ac.ir/article_50745_d90d55e08316779860740922b0388294.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
9
1
2018
07
01
Convergence and quantale-enriched categories
77
138
EN
Dirk
Hofmann
Center for Research and Development in Mathematics and Applications,
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.
dirk@ua.pt
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.
carla.reis@estgoh.ipc.pt
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
http://cgasa.sbu.ac.ir/article_58262.html
http://cgasa.sbu.ac.ir/article_58262_bab8553989f148c1daf7939ffd5b9f4d.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
9
1
2018
07
01
Convex $L$-lattice subgroups in $L$-ordered groups
139
161
EN
Rajabali
Borzooei
Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.
borzooei@sbu.ac.ir
Fateme
Hosseini
Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.
hoseini_nm@yahoo.com
Omid
Zahiri
University of Applied Science and Technology, Tehran, Iran
zahiri@protonmail.com
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
http://cgasa.sbu.ac.ir/article_50748.html
http://cgasa.sbu.ac.ir/article_50748_0ee3783313053dea8791d1990de4c8e2.pdf