2021-07-26T06:59:38Z
https://cgasa.sbu.ac.ir/?_action=export&rf=summon&issue=3075
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Cover for Vol. 6, No.1
2017
01
01
1
8
https://cgasa.sbu.ac.ir/article_41362_b680e243779384294d0a1a5f430d4e49.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Preface for Vol. 6, No.1
2017
01
01
1
2
https://cgasa.sbu.ac.ir/article_41363_632a529175edb41bcd68b95e5bd4129c.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
An interview with Bernhard Banaschewski
Christopher
Gilmour
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.
2017
01
01
11
16
https://cgasa.sbu.ac.ir/article_41364_f88e460e061873fa7fbb8a472837c522.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Everyday physics of extended bodies or why functionals need analyzing
F. William
Lawvere
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.
Functionals
physics
2017
01
01
9
19
https://cgasa.sbu.ac.ir/article_40434_9767cf7e5bb1b3e6e8b6cb442625ee0f.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Localic maps constructed from open and closed parts
Ales
Pultr
Jorge
Picado
Assembling a localic map $f\colon L\to M$ from localic maps $f_i\colon S_i\to M$, $i\in J$, 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.
frame
locale
sublocale
sublocale lattice
open sublocale
closed sublocale
localic map
preimage
Boolean frame
linear frame
2017
01
01
21
35
https://cgasa.sbu.ac.ir/article_15806_f90dc6ec251a402f3ff01305864296bd.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
The $lambda$-super socle of the ring of continuous functions
Simin
Mehran
Mehrdad
Namdari
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 $\lambda\leq |X|$) 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 $\lambda \geq\aleph_0$. 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 $x\in X$.
$lambda$-super socle
$lambda$-isolated point
one point $lambda$-compactification
$p_lambda$-space
2017
01
01
37
50
https://cgasa.sbu.ac.ir/article_33814_ae287573db032d67df112083dcb83c8f.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
C-connected frame congruences
Dharmanand
Baboolal
Paranjothi
Pillay
Ales
Pultr
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.
frame
frame congruence
congruence and sublocale lattice
connectedness
$T_D$-spatiality
2017
01
01
51
66
https://cgasa.sbu.ac.ir/article_34405_5a2102b74343052718d395077541ad72.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Slimming and regularization of cozero maps
Mohamad Mehdi
Ebrahimi
Abolghasem Karimi
Feizabadi
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 $c:M\rightarrow L$ in the general case where $A$ is not a ${\Bbb Q}$-algebra. We use the ring and module of fractions, in this construction process.
frame
cozero map
slim
slimming
algebraic
regular
Regularization
2017
01
01
67
84
https://cgasa.sbu.ac.ir/article_34407_a5c130e088026ede497dc3f85308de65.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Span and cospan representations of weak double categories
Marco
Grandis
Robert
Par'e
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.
Double category
tabulator
span
2017
01
01
85
105
https://cgasa.sbu.ac.ir/article_39606_b35c476a103000cd063c479026904e91.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
On MV-algebras of non-linear functions
Antonio
Di Nola
Giacomo
Lenzi
Gaetano
Vitale
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).
MV-algebra
McNaughton function
Hopfian algebra
2017
01
01
107
120
https://cgasa.sbu.ac.ir/article_40443_c18b8b2c7dd7e06904930c09a08f4a60.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Choice principles and lift lemmas
Marcel
Ern'e
We show that in ${\bf ZF}$ set theory without choice, the Ultrafilter 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 quasicontinuous 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.
Choice
(super)compact
foot
free semilattice
locale
noetherian
prime
sober
well-filtered
2017
01
01
121
146
https://cgasa.sbu.ac.ir/article_40448_f354e76a770fa82f66fa30955e1aba56.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2017
6
Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
Abstracts in Persian, Vol. 6, No. 1
2017
01
01
147
159
https://cgasa.sbu.ac.ir/article_41365_ca7d73e752dcdb282ba0f06142ad0069.pdf