Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701Representation of $H$-closed monoreflections in archimedean $ell$-groups with weak unit1136147510.29252/cgasa.9.1.1ENBernhardBanaschewskiDepartment of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada.Anthony W.HagerDepartment of Mathematics and CS, Wesleyan University, Middletown, CT 06459.Journal Article20170722 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 $C(T)$.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701Total graph of a $0$-distributive lattice15275074910.29252/cgasa.9.1.15ENShahabaddinEbrahimi AtaniFaculty of Mathematical Sciences, University of Guilan, Rasht, IranSabouraDolati PishhesariFaculty of Mathematical Sciences, University of Guilan, Rasht, IranMehdiKhoramdelFaculty of Mathematical Sciences, University of Guilan, Rasht, IranMaryamSedghiFaculty of Mathematical Sciences, University of Guilan, Rasht, IranJournal Article20170127Let £ 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701On lifting of biadjoints and lax algebras29585074710.29252/cgasa.9.1.29ENFernandoLucatelli NunesCMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal.Journal Article20170430Given 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701Pointfree topology version of image of real-valued continuous functions59755074510.29252/cgasa.9.1.59ENAbolghasemKarimi FeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.Ali AkbarEstajiFaculty of Mathematics and Computer Sciences,
Hakim Sabzevari University, Sabzevar, Iran.MaryamRobat SarpoushiFaculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.Journal Article20170318Let $ { 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$.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701Convergence and quantale-enriched categories771385826210.29252/cgasa.9.1.77ENDirkHofmannCenter for Research and Development in Mathematics and Applications,
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.CarlaD. ReisPolytechnic 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.Journal Article20170528Generalising 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701Convex $L$-lattice subgroups in $L$-ordered groups1391615074810.29252/cgasa.9.1.139ENRajabaliBorzooeiDepartment of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.FatemeHosseiniDepartment of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran.OmidZahiriUniversity of Applied Science and Technology, Tehran, IranJournal Article20170326In 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.