Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101Everyday physics of extended bodies or why functionals need analyzing91940434ENF. WilliamLawvereProfessor Emeritus, Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.Journal Article20161111 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101Localic maps constructed from open and closed parts213515806ENAlesPultrDepartment of Applied Mathematics and ITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.JorgePicadoCMUC, Department of Mathematics, University of Coimbra, Apar\-ta\-do 3008, 3001-501 Coimbra, Portugal.Journal Article20160502Assembling 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101The $lambda$-super socle of the ring of continuous functions375033814ENSiminMehranDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.MehrdadNamdariDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.Journal Article20160813The 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 $xin X$.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101C-connected frame congruences516634405ENDharmanandBaboolalSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.ParanjothiPillaySchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.AlesPultrDepartment of Applied Mathematics and CE-ITI, MFF, Charles University, Malostransk\'e n\'am. 24, 11800 Praha 1, Czech Republic.Journal Article20160601We 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101Slimming and regularization of cozero maps678434407ENMohamad MehdiEbrahimiDepartment of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.Abolghasem KarimiFeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.Journal Article20160527Cozero 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.<br /> Finally, we give slim regularization for a cozero map $c:Mrightarrow L$ in the general case where $A$ is not a ${Bbb Q}$-algebra. We use the ring and module of fractions, in this construction process.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101Span and cospan representations of weak double categories8510539606ENMarcoGrandisDipartimento di Matematica, Universit\`a di Genova, Via Dodecaneso 35,
16146-Genova, ItalyRobertPar\'eDepartment of Mathematics and Statistics, Dalhousie University,
Halifax NS, Canada B3H 4R2Journal Article20160822We 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101On MV-algebras of non-linear functions10712040443ENAntonioDi NolaDepartment of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.GiacomoLenziDepartment of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.GaetanoVitaleDepartment of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.Journal Article20160905In this paper, the main results are:<br />a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I;<br />a study of Hopfian MV-algebras; and<br />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).Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58536Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)20170101Choice principles and lift lemmas12114640448ENMarcelErn\'eFaculty for Mathematics and Physics, IAZD, Leibniz Universit\"at, Welfengarten 1, D 30167 Hannover, Germany.Journal Article20160929We 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.