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Categories and General Algebraic Structures with Applications - Volume:6 Issue:1, 2017
  • Volume:6 Issue:1, 2017
  • Special issue on the Occasion of Banaschewski’s 90th Birthday I
  • تاریخ انتشار: 1396/05/12
  • تعداد عناوین: 10
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  • Pages 1-2
  • Christopher Gilmour Pages 3-8
    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.
  • F. William Lawvere Pages 9-19
    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.
    Keywords: Functionals, physics
  • Ales Pultr, Jorge Picado Pages 21-35
    Assembling a localic map f:L→M from localic maps f i :S i →M i∈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.
    Keywords: Frame, locale, sublocale, sublocale lattice, open sublocale, closed sublocale, localic map, preimage, Boolean frame, linear frame
  • Simin Mehran, Mehrdad Namdari Pages 37-50
    he concept of λ-super socle of C(X), denoted by Sλ(X) (i.e., the set of elements of C(X) such that the cardinality of their cozerosets are less than λ, where λ is a regular cardinal number with λ≤|X|) is introduced and studied. Using this concept we extend some of the basic results concerning SCF(X), the super socle of C(X) to Sλ(X), where λ≥ℵ0. In particular, we determine spaces X for which SCF(X) and Sλ(X) coincide. The one-point λ-compactification of a discrete space is algebraically characterized via the concept of λ-super socle. In fact we show that X is the one-point λ-compactification of a discrete space Y if and only if Sλ(X) is a regular ideal and Sλ(X)=Ox, for some x∈X.
    Keywords: lambda-super socle, lambda-isolated point, one point lambda-compactification, plambda-space
  • Dharmanand Baboolal, Paranjothi Pillay, Ales Pultr Pages 51-66
    We discuss the congruences θ that are connected as elements of the (totally disconnected) congruence frame \CFL, 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 \CFL. They are, among others, shown to be TD -spatial; characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.
    Keywords: Frame, frame congruence, congruence, sublocale lattice, connectedness, TD-spatiality
  • Mohammad Mehdi Ebrahimi, Abolghasem Karimi Feizabadi Pages 67-84
    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 regularization for a cozero map c:M→L in the general case where A is not a Q -algebra. We use the ring and module of fractions, in this construction process.
    Keywords: Frame, cozero map, slim, slimming, algebraic, regular, regularization
  • Marco Grandis, Robert Pare Pages 85-105
    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.
    Keywords: Double category, tabulator, span
  • Antonio Di Nola, Giacomo Lenzi, Gaetano Vitale Pages 107-120
    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; and a 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).
    Keywords: MV-algebra, McNaughton function, Hopfian algebra
  • Marcel Erne Pages 121-146
    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 ZF
    set theory.
    Keywords: choice, (super)compact, fot, free semilattice, locale, noetherian, prime, sober, well-filtered