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