Categories and General Algebraic Structures with ApplicationsCategories and General Algebraic Structures with Applications
http://cgasa.sbu.ac.ir/
Mon, 22 Oct 2018 07:29:19 +0100FeedCreatorCategories and General Algebraic Structures with Applications
http://cgasa.sbu.ac.ir/
Feed provided by Categories and General Algebraic Structures with Applications. Click to visit.Cover for Vol. 9, No. 1.
http://cgasa.sbu.ac.ir/article_65930_8376.html
Sat, 30 Jun 2018 19:30:00 +0100Representation of $H$-closed monoreflections in archimedean $\ell$-groups with weak unit
http://cgasa.sbu.ac.ir/article_61475_8376.html
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 $C(T)$.Sat, 30 Jun 2018 19:30:00 +0100Total graph of a $0$-distributive lattice
http://cgasa.sbu.ac.ir/article_50749_8376.html
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.Sat, 30 Jun 2018 19:30:00 +0100On lifting of biadjoints and lax algebras
http://cgasa.sbu.ac.ir/article_50747_8376.html
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 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} 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 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.Sat, 30 Jun 2018 19:30:00 +0100Pointfree topology version of image of real-valued continuous functions
http://cgasa.sbu.ac.ir/article_50745_8376.html
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 $f:Xrightarrowmathbb R$ and $ S subseteq mathbb R$.Sat, 30 Jun 2018 19:30:00 +0100Convergence and quantale-enriched categories
http://cgasa.sbu.ac.ir/article_58262_8376.html
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.Sat, 30 Jun 2018 19:30:00 +0100Convex $L$-lattice subgroups in $L$-ordered groups
http://cgasa.sbu.ac.ir/article_50748_8376.html
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.Sat, 30 Jun 2018 19:30:00 +0100Persian Abstracts, Vol. 9, No. 1.
http://cgasa.sbu.ac.ir/article_65931_8376.html
Sat, 30 Jun 2018 19:30:00 +0100On the property $U$-($G$-$PWP$) of acts
http://cgasa.sbu.ac.ir/article_50746_0.html
In this paper first of all we introduce Property $U$-($G$-$PWP$) of acts, which is an extension of Condition $(G$-$PWP)$ and give some general properties. Then we give a characterization of monoids when this property of acts implies some others. Also we show that the strong (faithfulness, $P$-cyclicity) and ($P$-)regularity of acts imply the property $U$-($G$-$PWP$). Finally, we give a necessary and sufficient condition under which all (cyclic, finitely generated) right acts or all (strongly, $Re$-) torsion free (cyclic, finitely generated) right acts satisfy Property $U$-($G$-$PWP$).Tue, 03 Oct 2017 20:30:00 +0100State filters in state residuated lattices
http://cgasa.sbu.ac.ir/article_57443_0.html
In this paper, we introduce the notions of prime state filters, obstinate state filters, and primary state filters in state residuated lattices and study some properties of them. Several characterizations of these state filters are given and the prime state filter theorem is proved. In addition, we investigate the relations between them.Mon, 05 Feb 2018 20:30:00 +0100An equivalence functor between local vector lattices and vector lattices
http://cgasa.sbu.ac.ir/article_61405_0.html
We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-trivial components. Nevertheless, our main purpose is to prove, via what we call the radical functor, that the category of all vector lattices and lattice homomorphisms is equivalent to the category of local vectors lattices and unital (i.e., unit preserving) lattice homomorphisms.Mon, 30 Apr 2018 19:30:00 +0100Lattice of compactifications of a topological group
http://cgasa.sbu.ac.ir/article_61406_0.html
We show that the lattice of compactifications of a topological group $G$ is a complete lattice which is isomorphic to the lattice of all closed normal subgroups of the Bohr compactification $bG$ of $G$. The correspondence defines a contravariant functor from the category of topological groups to the category of complete lattices. Some properties of the compactification lattice of a topological group are obtained.Mon, 30 Apr 2018 19:30:00 +0100Mappings to Realcompactifications
http://cgasa.sbu.ac.ir/article_61474_0.html
In this paper, we introduce and study a mapping from the collection of all intermediate rings of $C(X)$ to the collection of all realcompactifications of $X$ contained in $beta X$. By establishing the relations between this mapping and its converse, we give a different approach to the main statements of De et. al. Using these, we provide different answers to the four basic questions raised in Acharyya et.al. Finally, we give some notes on the realcompactifications generated by ideals.Thu, 03 May 2018 19:30:00 +0100A Universal Investigation of $n$-representations of $n$-quivers
http://cgasa.sbu.ac.ir/article_63576_0.html
noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$, called an $n$-quiver, and identify each category $Rep_k(mathit{Q}_j)$ of representations of a quiver $mathit{Q}_j$ as a full subcategory of the category $Rep_k(mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}})$ of representations of $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$ for every $j in {1,2,ldots , n}$.Sun, 10 Jun 2018 19:30:00 +0100The category of generalized crossed modules
http://cgasa.sbu.ac.ir/article_69897_0.html
In the definition of a crossed module $(T,G,rho)$, the actions of the group $T$ and $G$ on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category ${bf GCM}$, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate some of its categorical properties. In particular, we study the relations between epimorphisms and the surjective morphisms, and thus generalize the corresponding results of the category of (ordinary) crossed modules. By generalizing the conjugation action, we can find out what is the superiority of the conjugation to other actions. Also, we can find out a generalized crossed module with which other actions (other than the conjugation) has the properties same as a crossed module.Fri, 28 Sep 2018 20:30:00 +0100