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 $C$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in C$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar classical counterpart concerning {it finite closed} covers of a space $X$ (Picado and Pultr cite{4}). This note presents alternative proofs of these pointfree results which differ from those of cite{4} by treating the issue in terms of {it frame homomorphisms} while the latter deals with the dual situation concerning {it 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. 6, No.1
http://www.cgasa.ir/article_41362_3074.html
Sat, 31 Dec 2016 20:30:00 +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 $T(M)$ 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 $M/T(M)cong M_lambda/T(M_lambda)$. 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. 6, No.1
http://www.cgasa.ir/article_41363_3074.html
Sat, 31 Dec 2016 20:30:00 +0100$\mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps
http://www.cgasa.ir/article_38548_0.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, 28 Oct 2016 20:30:00 +0100An interview with Bernhard Banaschewski
http://www.cgasa.ir/article_41364_3074.html
This interview, a co-operative eort of Bernhard Banaschewski and Christopher Gilmour, took place over a few days in December, 2016. It was nalised over coee and a shared slice of excellent cheesecake at The Botanical Tea Garden, a small, home situated, tea garden in Little Mowbray, Cape Town.Wed, 30 Nov 2016 20:30:00 +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 +0100Everyday physics of extended bodies or why functionals need analyzing
http://www.cgasa.ir/article_40434_3074.html
Functionals were discovered and used by Volterra over a century ago in his study of the motions of viscous elastic materials and electromagnetic fields. The need to precisely account for the qualitative effects of the cohesion and shape of the domains of these functionals was the major impetus to the development of the branch of mathematics known as topology, and today large numbers of mathematicians still devote their work to a detailed technical analysis of functionals. Yet the concept needs to be understood by all people who want to fully participate in 21st century society. Through some explicit use of mathematical categories and their transformations, functionals can be treated in a way which is non-technical and yet permits considerable reliable development of thought. We show how a deformable body such as a storm cloud can be viewed as a kind of space in its own right, as can an interval of time such as an afternoon; the infinite-dimensional spaces of configurations of the body and of its states of motion are constructed, and the role of the infinitesimal law of its motion revealed. We take nilpotent infinitesimals as given, and follow Euler in defining real numbers as ratios of infinitesimals.Wed, 30 Nov 2016 20:30:00 +0100Localic maps constructed from open and closed parts
http://www.cgasa.ir/article_15806_3074.html
Assembling a localic map $fcolon Lto M$ from localic maps $f_icolon S_ito M$, $iin J$, defined on closed resp. open sublocales $(J$ finite in the closed case$)$ follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead, we present simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper.Wed, 30 Nov 2016 20:30:00 +0100Some Types of Filters in Equality Algebras (dedicated to BB 90th birthday)
http://www.cgasa.ir/article_42342_0.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.Tue, 28 Feb 2017 20:30:00 +0100The $\lambda$-super socle of the ring of continuous functions
http://www.cgasa.ir/article_33814_3074.html
The concept of $lambda$-super socle of $C(X)$, denoted by $S_lambda(X)$ (i.e., the set of elements of $C(X)$ such that the cardinality of their cozerosets are less than $lambda$, where $lambda$ is a regular cardinal number with $lambdaleq |X|$) is introduced and studied. Using this concept we extend some of the basic results concerning $SC_F(X)$, the super socle of $C(X)$ to $S_lambda(X)$, where $lambda geqaleph_0$. In particular, we determine spaces $X$ for which $SC_F(X)$ and $S_lambda(X)$ coincide. The one-point $lambda$-compactification of a discrete space is algebraically characterized via the concept of $lambda$-super socle. In fact we show that $X$ is the one-point $lambda$-compactification of a discrete space $Y$ if and only if $S_lambda(X)$ is a regular ideal and $S_lambda(X)=O_x$, for some $xin X$.Wed, 30 Nov 2016 20:30:00 +0100Tangled Closure Algebras (dedicated to BB 90th birthday)
http://www.cgasa.ir/article_42354_0.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.Wed, 25 Jan 2017 20:30:00 +0100C-connected frame congruences
http://www.cgasa.ir/article_34405_3074.html
We discuss the congruences $theta$ that are connected as elements of the (totally disconnected) congruence frame $CF L$, and show that they are in a one-to-one correspondence with the completely prime elements of $L$, giving an explicit formula. Then we investigate those frames $L$ with enough connected congruences to cover the whole of $CF L$. They are, among others, shown to be $T_D$-spatial; characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.Wed, 30 Nov 2016 20:30:00 +0100One-point compactifications and continuity for partial frames (dedicated to BB 90th birthday)
http://www.cgasa.ir/article_43180_0.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.Sun, 12 Feb 2017 20:30:00 +0100Slimming and regularization of cozero maps
http://www.cgasa.ir/article_34407_3074.html
Cozero maps are generalized forms of cozero elements. Two particular cases of cozero maps, slim and regular cozero maps, are significant. In this paper we present methods to construct slim and regular cozero maps from a given cozero map. The construction of the slim and the regular cozero map from a cozero map are called slimming and regularization of the cozero map, respectively. Also, we prove that the slimming and regularization create reflector functors, and so we may say that they are the best method of constructing slim and regular cozero maps, in the sense of category theory. Finally, we give slim regularizationfor a cozero map $c:Mrightarrow L$ in the general case where $A$is not a ${Bbb Q}$-algebra. We use the ring and module offractions, in this construction process.Wed, 30 Nov 2016 20:30:00 +0100Adjoint relations for the category of local dcpos (dedicated to BB 90th birthday)
http://www.cgasa.ir/article_43374_0.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.Tue, 28 Feb 2017 20:30:00 +0100Span and cospan representations of weak double categories
http://www.cgasa.ir/article_39606_3074.html
We prove that many important weak double categories can be `represented' by spans, using the basic higher limit of the theory: the tabulator. Dually, representations by cospans via cotabulators are also frequent.Wed, 30 Nov 2016 20:30:00 +0100On MV-algebras of non-linear functions
http://www.cgasa.ir/article_40443_3074.html
In this paper, the main results are:a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I;a study of Hopfian MV-algebras; anda category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism).Wed, 30 Nov 2016 20:30:00 +0100Choice principles and lift lemmas
http://www.cgasa.ir/article_40448_3074.html
We show that in {bf ZF} set theory without choice, the Ultrafilter mbox{Principle} ({bf UP}) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasi-continuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from {bf UP} but also from {bf DC}, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ${bf ZF}$ set theory.Wed, 30 Nov 2016 20:30:00 +0100Abstracts in Persian, Vol. 6, No. 1
http://www.cgasa.ir/article_41365_3074.html
Sat, 31 Dec 2016 20:30:00 +0100