Categories and General Algebraic Structures with ApplicationsCategories and General Algebraic Structures with Applications
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Feed provided by Categories and General Algebraic Structures with Applications. Click to visit.On the pointfree counterpart of the local definition of classical continuous maps
http://www.cgasa.ir/article_32712_0.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.Fri, 09 Sep 2016 19:30:00 +0100Cover for Vol. 7, No.1
http://www.cgasa.ir/article_48553_5757.html
Fri, 30 Jun 2017 19:30:00 +0100On finitely generated modules whose first nonzero Fitting ideals are regular
http://www.cgasa.ir/article_33815_0.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.Mon, 03 Oct 2016 20:30:00 +0100Preface for Vol. 7, No.1
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Fri, 30 Jun 2017 19:30:00 +0100An interview with Bernhard Banaschewski
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This interview, a co-operative effort of Bernhard Banaschewski and Christopher Gilmour, took place over a few days in December, 2016. It was finalised over coffee and a shared slice of excellent cheesecake at The Botanical Tea Garden, a small, home situated, tea garden in Little Mowbray, Cape Town.Fri, 30 Jun 2017 19:30:00 +0100Equivalences in Bicategories
http://www.cgasa.ir/article_39393_0.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.Sat, 12 Nov 2016 20:30:00 +0100Tangled Closure Algebras
http://www.cgasa.ir/article_42354_5757.html
The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical `tangle modality' connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.Fri, 30 Jun 2017 19:30:00 +0100Some Types of Filters in Equality Algebras
http://www.cgasa.ir/article_42342_5757.html
Equality algebras were introduced by S. Jenei as a possible algebraic semantic for fuzzy type theory. In this paper, we introduce some types of filters such as (positive) implicative, fantastic, Boolean, and prime filters in equality algebras and we prove some results which determine the relation between these filters. We prove that the quotient equality algebra induced by an implicative filter is a Boolean algebra, by a fantastic filter is a commutative equality algebra, and by a prime filter is a chain, under suitable conditions. Finally, we show that positive implicative, implicative, and Boolean filters are equivalent on bounded commutative equality algebras.Fri, 30 Jun 2017 19:30:00 +0100One-point compactifications and continuity for partial frames
http://www.cgasa.ir/article_43180_5757.html
Locally compact Hausdorff spaces and their one-point compactifications are much used in topology and analysis; in lattice and domain theory, the notion of continuity captures the idea of local compactness. Our work is located in the setting of pointfree topology, where lattice-theoretic methods can be used to obtain topological results.Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such.Partial frames are meet-semilattices in which not all subsets need have joins.A distinguishing feature of their study is that a small collection of axioms of an elementary nature allows one to do much that is traditional for frames or locales. The axioms are sufficiently general to include as examples $sigma$-frames, $kappa$-frames and frames.In this paper, we present the notion of a continuous partial frame by means of a suitable ``way-below'' relation; in the regular case this relation can be characterized using separating elements, thus avoiding any use of pseudocomplements (which need not exist in a partial frame). Our first main result is an explicit construction of a one-point compactification for a regular continuous partial frame using generators and relations. We use strong inclusions to link continuity and one-point compactifications to least compactifications. As an application, we show that a one-point compactification of a zero-dimensional continuous partial frame is again zero-dimensional. We next consider arbitrary compactifications of regular continuous partial frames. In full frames, the natural tools to use are right and left adjoints of frame maps; in partial frames these are, in general, not available. This necessitates significantly different techniques to obtain largest and smallest elements of fibres (which we call balloons); these elements are then used to investigate the structure of the compactifications. We note that strongly regular ideals play an important r^{o}le here. The paper concludes with a proof of the uniqueness of the one-point compactification.Fri, 30 Jun 2017 19:30:00 +0100Adjoint relations for the category of local dcpos
http://www.cgasa.ir/article_43374_5757.html
In this paper, we consider the forgetful functor from the category {bf LDcpo} of local dcpos (respectively, {bf Dcpo} of dcpos) to the category {bf Pos} of posets (respectively, {bf LDcpo} of local dcpos), and study the existence of its left and right adjoints. Moreover, we give the concrete forms of free and cofree $S$-ldcpos over a local dcpo, where $S$ is a local dcpo monoid. The main results are: (1) The forgetful functor $U$ : {bf LDcpo} $longrightarrow$ {bf Pos} has a left adjoint, but does not have a right adjoint;(2) The inclusion functor $I$ : {bf Dcpo} $longrightarrow$ {bf LDcpo} has a left adjoint, but does not have a right adjoint;(3) The forgetful functor $U$ : {bf LDcpo}-$S$ $longrightarrow$ {bf LDcpo} hasboth left and right adjoints;(4) If $(S,cdot,1)$ is a good ldcpo-monoid, then the forgetful functor $U$: {bf LDcpo}-$S$ $longrightarrow$ {bf Pos}-$S$ has a left adjoint.Fri, 30 Jun 2017 19:30:00 +0100Filters of Coz(X)
http://www.cgasa.ir/article_44925_5757.html
In this article we investigate filters of cozero sets for real-valued continuous functions, called $coz$-filters. Much is known for $z$-ultrafilters and their correspondence with maximal ideals of $C(X)$. Similarly, a correspondence will be established between $coz$-ultrafilters and minimal prime ideals of $C(X)$. We will further notice various properties of $coz$-ultrafilters in relation to $P$-spaces and $F$-spaces. In the last two sections, the collection of $coz$-ultrafilters will be topologized, and then compared to the hull-kernel and the inverse topologies placed on the collection of minimal prime ideals of $C(X)$ and general lattice-ordered groups.Fri, 30 Jun 2017 19:30:00 +0100On (po-)torsion free and principally weakly (po-)flat $S$-posets
http://www.cgasa.ir/article_44578_0.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.Thu, 30 Mar 2017 19:30:00 +0100Perfect secure domination in graphs
http://www.cgasa.ir/article_44926_5757.html
Let $G=(V,E)$ be a graph. A subset $S$ of $V$ is a dominating set of $G$ if every vertex in $Vsetminus S$ is adjacent to a vertex in $S.$ A dominating set $S$ is called a secure dominating set if for each $vin Vsetminus S$ there exists $uin S$ such that $v$ is adjacent to $u$ and $S_1=(Ssetminus{u})cup {v}$ is a dominating set. If further the vertex $uin S$ is unique, then $S$ is called a perfect secure dominating set. The minimum cardinality of a perfect secure dominating set of $G$ is called the perfect secure domination number of $G$ and is denoted by $gamma_{ps}(G).$ In this paper we initiate a study of this parameter and present several basic results.Fri, 30 Jun 2017 19:30:00 +0100$\mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps
http://www.cgasa.ir/article_38548_5757.html
In this paper, for each {it lattice-valued map} $Arightarrow L$ with some properties, a ring representation $Arightarrow mathcal{R}L$ is constructed. This representation is denoted by $tau_c$ which is an $f$-ring homomorphism and a $mathbb Q$-linear map, where its index $c$, mentions to a lattice-valued map. We use the notation $delta_{pq}^{a}=(a -p)^{+}wedge (q-a)^{+}$, where $p, qin mathbb Q$ and $ain A$, that is nominated as {it interval projection}. To get a well-defined $f$-ring homomorphism $tau_c$, we need such concepts as {it bounded}, {it continuous}, and $mathbb Q$-{it compatible} for $c$, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued map $c_{phi}:Arightarrow L $ for each $f$-ring homomorphism $phi: Arightarrow mathcal{R}L$. It is proved that $c_{tau_c}=c^r$ and $tau_{c_{phi}}=phi$, which they make a kind of correspondence relation between ring representations $Arightarrow mathcal{R}L$ and the lattice-valued maps $Arightarrow L$, Where the mapping $c^r:Arightarrow L$ is called a {it realization} of $c$. It is shown that $tau_{c^r}=tau_c$ and $c^{rr}=c^r$. Finally, we describe how $tau_c$ can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski. Fri, 30 Jun 2017 19:30:00 +0100The projectable hull of an archimedean $\ell$-group with weak unit
http://www.cgasa.ir/article_46629_5757.html
The much-studied projectable hull of an $ell$-group $Gleq pG$ is an essential extension, so that, in the case that $G$ is archimedean with weak unit, ``$Gin {bf W}$", we have for the Yosida representation spaces a ``covering map" $YG leftarrow YpG$. We have earlier cite{hkm2} shown that (1) this cover has a characteristic minimality property, and that (2) knowing $YpG$, one can write down $pG$. We now show directly that for $mathscr{A}$, the boolean algebra in the power set of the minimal prime spectrum $Min(G)$, generated by the sets $U(g)={Pin Min(G):gnotin P}$ ($gin G$), the Stone space $mathcal{A}mathscr{A}$ is a cover of $YG$ with the minimal property of (1); this extends the result from cite{bmmp} for the strong unit case. Then, applying (2) gives the pre-existing description of $pG$, which includes the strong unit description of cite{bmmp}. The present methods are largely topological, involving details of covering maps and Stone duality.Fri, 30 Jun 2017 19:30:00 +0100Persian Abstracts
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Fri, 30 Jun 2017 19:30:00 +0100A note on the problem when FS-domains coincide with RB-domains
http://www.cgasa.ir/article_47217_0.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.Tue, 20 Jun 2017 19:30:00 +0100On Property (A) and the socle of the $f$-ring $Frm(\mathcal{P}(\mathbb R), L)$
http://www.cgasa.ir/article_49786_0.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).Thu, 31 Aug 2017 19:30:00 +0100Pointfree topology version of image of real-valued continuous functions
http://www.cgasa.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://www.cgasa.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
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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://www.cgasa.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://www.cgasa.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 +0100