Categories and General Algebraic Structures with ApplicationsCategories and General Algebraic Structures with Applications
http://cgasa.sbu.ac.ir/
Tue, 24 Apr 2018 01:47:13 +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. 8, No. 1.
http://cgasa.sbu.ac.ir/article_55242_6845.html
.Sun, 31 Dec 2017 20:30:00 +0100On the pointfree counterpart of the local definition of classical continuous maps
http://cgasa.sbu.ac.ir/article_32712_6845.html
The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $varphi_U: U to Y$ on each member $U$ of an open cover ${mathfrak C}$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in {mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar classical counterpart concerning finite closed covers of a space $X$ (Picado and Pultr [4]). This note presents alternative proofs of these pointfree results which differ from those of [4] by treating the issue in terms of frame homomorphisms while the latter deals with the dual situation concerning localic maps. A notable advantage of the present approach is that it also provides proofs of the analogous results for some significant variants of frames which are not covered by the localic arguments.Sun, 31 Dec 2017 20:30:00 +0100On finitely generated modules whose first nonzero Fitting ideals are regular
http://cgasa.sbu.ac.ir/article_33815_6845.html
A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of $R^n$ which is generated by columns of a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set. Let $M=R^n/N$ be a module of type ($F_{n-1}$) and ${rm T}(M)$ be the submodule of $M$ consisting of all elements of $M$ that are annihilated by a regular element of $R$. For $ lambdain Lambda $, put $M_lambda=R^n/<(a_{1lambda},...,a_{nlambda})^t>$. The main result of this paper asserts that if $M_lambda $ is a regular $R$-module, for some $lambdainLambda$, then $M/{rm T}(M)cong M_lambda/{rm T}(M_lambda)$. Also it is shown that if $M_lambda$ is a regular torsionfree $R$-module, for some $lambdain Lambda$, then $ Mcong M_lambda. $ As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.Sun, 31 Dec 2017 20:30:00 +0100Equivalences in Bicategories
http://cgasa.sbu.ac.ir/article_39393_6845.html
In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In particular, all calculations done in a bicategory are fully explicit, in order to overcome the difficulties which arise when working with bicategories instead of 2-categories.Sun, 31 Dec 2017 20:30:00 +0100On (po-)torsion free and principally weakly (po-)flat $S$-posets
http://cgasa.sbu.ac.ir/article_44578_6845.html
In this paper, we first consider (po-)torsion free and principally weakly (po-)flat $S$-posets, specifically we discuss when (po-)torsion freeness implies principal weak (po-)flatness. Furthermore, we give a counterexample to show that Theorem 3.22 of Shi is incorrect. Thereby we present a correct version of this theorem. Finally, we characterize pomonoids over which all cyclic $S$-posets are weakly po-flat.Sun, 31 Dec 2017 20:30:00 +0100A note on the problem when FS-domains coincide with RB-domains
http://cgasa.sbu.ac.ir/article_47217_6845.html
In this paper, we introduce the notion of super finitely separating functions which gives a characterization of RB-domains. Then we prove that FS-domains and RB-domains are equivalent in some special cases by the following three claims: a dcpo is an RB-domain if and only if there exists an approximate identity for it consisting of super finitely separating functions; a consistent join-semilattice is an FS-domain if and only if it is an RB-domain; an L-domain is an FS-domain if and only if it is an RB-domain. These results are expected to provide useful hints to the open problem of whether FS-domains are identical with RB-domains.Sun, 31 Dec 2017 20:30:00 +0100On Property (A) and the socle of the $f$-ring $Frm(\mathcal{P}(\mathbb R), L)$
http://cgasa.sbu.ac.ir/article_49786_6845.html
For a frame $L$, consider the $f$-ring $ mathcal{F}_{mathcal P}L=Frm(mathcal{P}(mathbb R), L)$. In this paper, first we show that each minimal ideal of $ mathcal{F}_{mathcal P}L$ is a principal ideal generated by $f_a$, where $a$ is an atom of $L$. Then we show that if $L$ is an $mathcal{F}_{mathcal P}$-completely regular frame, then the socle of $ mathcal{F}_{mathcal P}L$ consists of those $f$ for which $coz (f)$ is a join of finitely many atoms. Also it is shown that not only $ mathcal{F}_{mathcal P}L$ has Property (A) but also if $L$ has a finite number of atoms then the residue class ring $ mathcal{F}_{mathcal P}L/mathrm{Soc}( mathcal{F}_{mathcal P}L)$ has Property (A).Sun, 31 Dec 2017 20:30:00 +0100Persian Abstracts, Vol. 8.
http://cgasa.sbu.ac.ir/article_55243_6845.html
.Sun, 31 Dec 2017 20:30:00 +0100Pointfree topology version of image of real-valued continuous functions
http://cgasa.sbu.ac.ir/article_50745_0.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$.Tue, 03 Oct 2017 20:30:00 +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 +0100On lifting of biadjoints and lax algebras
http://cgasa.sbu.ac.ir/article_50747_0.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.Tue, 03 Oct 2017 20:30:00 +0100Convex $L$-lattice subgroups in $L$-ordered groups
http://cgasa.sbu.ac.ir/article_50748_0.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.Tue, 03 Oct 2017 20:30:00 +0100Total graph of a $0$-distributive lattice
http://cgasa.sbu.ac.ir/article_50749_0.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.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 +0100Convergence and quantale-enriched categories
http://cgasa.sbu.ac.ir/article_58262_0.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, 03 Mar 2018 20:30:00 +0100