Quaestiones Mathematicae https://www.ajol.info/index.php/qm <p><em>Quaestiones Mathematicae</em> is devoted to research articles from a wide range of mathematical areas. Longer expository papers of exceptional quality are also considered. Published in English, the journal receives contributions from authors around the globe and serves as an important reference source for anyone interested in mathematics.</p> <p>Read more about the journal <a href="http://www.nisc.co.za/products/12/journals/quaestiones-mathematicae" target="_blank" rel="noopener">here</a>.&nbsp;</p> Taylor & Francis en-US Quaestiones Mathematicae 1607-3606 Copyright for articles published in this journal is retained by the journal. Special issue dedicated to the memory of Professor Horst Herrlich guest editors’ introduction https://www.ajol.info/index.php/qm/article/view/260204 <p>No Abstract</p> Klaas Pieter Hart Mirek Husek Jan van Mill Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 1 1 Horst Herrlich – a deep and enduring Contribution to South African Mathematics https://www.ajol.info/index.php/qm/article/view/260210 <p>No Abstract</p> David Holgate Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 3 11 Smallness in topology https://www.ajol.info/index.php/qm/article/view/260214 <p>Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy&nbsp; theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological&nbsp; spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many&nbsp; familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant&nbsp; differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition&nbsp; standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective&nbsp; subcategories of the category of all topological spaces.&nbsp;</p> Jiri Adamek Miroslav Husek Jiri Rosicky Walter Tholen Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 13 39 Some concepts of topology and questions in topological algebra https://www.ajol.info/index.php/qm/article/view/260217 <p>This paper has features of mixed nature. It contains new results, in particular, on semitopological groups. We also pose some problems,&nbsp; introduce new definitions and describe in details certain techniques we need below, providing the proofs of not so well-known theorems&nbsp; for the sake of completeness. Hence, this article can be treated also as a kind of a short survey.&nbsp;&nbsp;</p> A.V. Arhangela'skii Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 13–39 13–39 The ℓP-metrization of functors with finite supports https://www.ajol.info/index.php/qm/article/view/260227 <p>No Abstract<br><br></p> Taras Banakh Viktoria Brydun Lesia Karchevska Mykhailo Zarichnyi Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 53 144 Lax comma categories of ordered sets https://www.ajol.info/index.php/qm/article/view/260244 <p>Let Ord be the category of (pre)ordered sets. Unlike Ord{X, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category Ord//X. In this paper we show that the forgetful functor Ord//X Ñ Ord is topological if and only if X is&nbsp; complete. Moreover, under suitable hypothesis, Ord//X is complete and cartesian closed if and only if X is. We end by analysing descent in&nbsp; this category. Namely, when X is complete, we show that, for a morphism in Ord//X, being pointwise effective for descent in Ord is&nbsp; sufficient, while being effective for descent in Ord is necessary, to be effective for descent in Ord//X.&nbsp;&nbsp;</p> Maria Manuel Clementino Fernando Lucatelli Nunes Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 145–159 145–159 Supported approach spaces https://www.ajol.info/index.php/qm/article/view/260246 <p>In this paper we work in the category of approach spaces with contractions [14], the objects of which are sets endowed with a numerical distance between sets and points. Approach spaces are to be considered a simultaneous generalization of both quasi-metric and&nbsp; topological spaces. Especially the fundamental notion of distance is reminiscent of the closure operator in a topological space and of the point-to-set distance in a quasi-metric space. The embedding of the category of topological spaces with continuous maps and of quasi- metric spaces with non-expansive maps is extremely nice. Every approach space has both a quasi-metric coreflection as well as a&nbsp; topological coreflection. Different approach spaces though can have the same topological as well as the same quasi-metric coreflection,&nbsp; in other words, in general these coreflections do not determine the approach space. In this paper we investigate&nbsp; approach spaces for which these coreflections, do determine the approach space. We will call such spaces supported. We prove that in the setting of compact approach spaces many examples of supported approach spaces can be found. Thus, compact spaces that are&nbsp; base-regular, which is a weakening of regularity, are always supported. An important feature of supported approach spaces is the&nbsp; behaviour of contractions. On a supported domain contractivity is characterized by the combination of continuity for the topological&nbsp; coreflection and non-expansiveness for the quasi-metric coreflection. This result implies that a supported approach space actually is the&nbsp; infimum of its quasi-metric and its topological coreflection. In the course of our study we also give several more examples of both supported and non-supported approach spaces.</p> E. Colebunders R. Lowen Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 161–190 161–190 Epimorphisms and closure operators of categories of semilattices https://www.ajol.info/index.php/qm/article/view/260247 <p>Motivated by a problem posed in [10], we investigate the closure operators of the category SLatt of join semilattices and its subcategory SLatt0 of join semilattices with bottom element. In particular, we show that there are only finitely many closure operators of both&nbsp; categories, and provide a complete classification. We use this result to deduce the known fact that epimorphisms of SLatt and SLatt0 are surjective. We complement the paper with two different proofs of this result using either generators or Isbell’s zigzag theorem.&nbsp;&nbsp;&nbsp;</p> D. Dikranjan A. Giordano Bruno N. Zava Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 191–221 191–221 Topogenous orders and related families of morphisms https://www.ajol.info/index.php/qm/article/view/260248 <p>In a category C with a proper (E,M)-factorization system, we study the notions of strict, co-strict, initial and final morphisms with respect&nbsp; to a topogenous order. Besides showing that they allow simultaneous study of four classes of morphisms obtained separately with&nbsp; respect to closure, interior and neighbourhood operators, the initial and final morphisms lead us to the study of topogenous orders&nbsp; induced by pointed and co-pointed endofunctors. We also lift the topogenous orders along an M-fibration. This permits one to obtain the&nbsp; lifting of interior and neighbourhood operators along an M-fibration and includes the lifting of closure operators found in the literature.&nbsp; A number of examples presented at the end of the paper demonstrates our results.</p> David Holgate Minani Iragi Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 223–238 223–238 Some properties of conjunctivity (subfitness) in generalized settings https://www.ajol.info/index.php/qm/article/view/260250 <p>The property of subfitness used in point-free topology (roughly speaking) to replace the slightly stronger T1-separation, appeared (as&nbsp; disjunctivity) already in the pioneering Wallman’s [16], then practically disappeared to reappear again (conjunctivity, subfitness), until it&nbsp; was in the recent decades recognized as an utmost important condition playing a very special role. Recently, it was also observed that this property (or its dual) appeared independently in general poset setting (e.g. as separativity in connection with forcing). In a recent&nbsp; paper [2], Delzell, Ighedo and Madden discussed it in the context of semilattices. In this article we discuss it on the background of the&nbsp; systems of meet-sets (subsets closed under existing infima) in posets of various generality (semilattices, lattices, distributive lattices,&nbsp; complete lattices) and present parallels of some localic (frame) facts, including a generalized variant of fitness.&nbsp;</p> M. Andrew Moshier Jorge Picado Ales Pultr Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 239–252 239–252 The monoidal nature of the Feistel-Toffoli construction https://www.ajol.info/index.php/qm/article/view/260251 <p>The Feistel-Toffoli construction of a bijective Boolean function out of an arbitrary one, a fundamental tool in reversible computing and in&nbsp; cryptography, has recently been analyzed (see [12]) to be a special instance of the construction of a monoid homomorphism from the&nbsp; X-fold cartesian power of a monoid M into the endomorphism monoid of the free M-set over the set X. It is the purpose of this note to&nbsp; show that this construction itself is in fact a genuine monoidal one. The generalization of the Feistel-Toffoli construction to internal&nbsp; categories in arbitrary finitely complete categories of [12] then becomes a special instance of this monoidal description.</p> Hans-E. Porst Copyright (c) 2023 2023-11-29 2023-11-29 46 S1 253–265 253–265