Recent zbMATH articles in MSC 06Bhttps://zbmath.org/atom/cc/06B2023-05-08T18:47:08.967005ZWerkzeugDistributive and completely distributive lattice extensions of ordered setshttps://zbmath.org/1507.060052023-05-08T18:47:08.967005Z"Morton, W."https://zbmath.org/authors/?q=ai:morton.wilmari"van Alten, Clint J."https://zbmath.org/authors/?q=ai:van-alten.clint-jSummary: It is known that a poset can be embedded into a distributive lattice if, and only if, it satisfies the prime filter separation property. We describe here a class of ``prime filter completions'' for posets with the prime filter separation property that are completely distributive lattices generated by the poset and preserve existing finite meets and joins. The free completely distributive lattice generated by a poset can be obtained through such a prime filter completion. We also show that every completely distributive completion of a poset with the prime filter separation property is representable as a canonical extension of the poset with respect to some set of filters and ideals. The connections between the prime filter completions and canonical extensions are described and yield the following corollary: the canonical extension of any distributive lattice is the free completely distributive lattice generated by the lattice. A construction that is a variant of the prime filter completion is given that can be used to obtain the free distributive lattice generated by a poset. In addition, it is shown that every distributive lattice extension of the poset can be represented by such a construction. Finally, we show that a poset with the prime filter separation property and the free distributive lattice generated by it generates the same free completely distributive lattice.Congruences on near-Heyting algebrashttps://zbmath.org/1507.060062023-05-08T18:47:08.967005Z"González, Luciano J."https://zbmath.org/authors/?q=ai:gonzalez.luciano-javier"Lattanzi, Marina B."https://zbmath.org/authors/?q=ai:lattanzi.marina-beatrizSummary: A near-Heyting algebra is a join-semilattice with a top element such that every principal upset is a Heyting algebra. We establish a one-to-one correspondence between the lattices of filters and congruences of a near-Heyting algebra. To attain this aim, we first show an embedding from the lattice of filters to the lattice of congruences of a distributive nearlattice. Then, we describe the subdirectly irreducible and simple near-Heyting algebras. Finally, we fully characterize the principal congruences of distributive nearlattices and near-Heyting algebras. We conclude that the varieties of distributive nearlattices and near-Heyting algebras have equationally definable principal congruences.Some preliminary results on the set of principal congruences of a finite latticehttps://zbmath.org/1507.060092023-05-08T18:47:08.967005Z"Grätzer, G."https://zbmath.org/authors/?q=ai:gratzer.george"Lakser, H."https://zbmath.org/authors/?q=ai:lakser.harrySummary: In [the first author, The congruences of a finite lattice. A ``proof-by-picture'' approach. 2nd edition. Basel: Birkhäuser/Springer (2016; Zbl 1348.06001)], Problem 22.1 asks for a characterization of subsets \(Q\) of a finite distributive lattice \(D\) such that there is a finite lattice \(L\) whose congruence lattice is isomorphic to \(D\) and under this isomorphism \(Q\) corresponds the the principal congruences of \(L\). In this note, we prove some preliminary results.A new dcpo whose Scott topology is well-filtered but not soberhttps://zbmath.org/1507.060102023-05-08T18:47:08.967005Z"Zhao, Dongsheng"https://zbmath.org/authors/?q=ai:zhao.dongsheng"Xi, Xiaoyong"https://zbmath.org/authors/?q=ai:xi.xiaoyong"Chen, Yixiang"https://zbmath.org/authors/?q=ai:chen.yixiangSummary: A topological space is sober if each of its nonempty closed irreducible sets is the closure of a unique singleton. A weaker topological property is the well-filteredness first considered by Heckmann. Johnstone constructed the first directed complete poset whose Scott topology is not sober. Heckmann asked whether the Scott topology of a directed complete poset is sober if it is assumed to be well-filtered. Kou constructed the first counterexample to give a negative answer. In this short note, based on our recent work on the dcpo models of topological space, we give another simpler example of directed complete poset whose Scott topology is well-filtered but not sober.Congruence structure of planar semimodular lattices: the general swing lemmahttps://zbmath.org/1507.060122023-05-08T18:47:08.967005Z"Czédli, Gábor"https://zbmath.org/authors/?q=ai:czedli.gabor"Grätzer, George"https://zbmath.org/authors/?q=ai:gratzer.george"Lakser, Harry"https://zbmath.org/authors/?q=ai:lakser.harrySummary: The swing lemma, proved by \textit{G. Grätzer} [Acta Sci. Math. 81, No. 3--4, 381--397 (2015; Zbl 1374.06017)], describes how a congruence spreads from a prime interval to another in a slim (having no \(\mathrm {M}_{3}\) sublattice), planar, semimodular lattice. We generalize the swing lemma to planar semimodular lattices.Embedding locales and formal topologies into positive topologieshttps://zbmath.org/1507.060172023-05-08T18:47:08.967005Z"Ciraulo, Francesco"https://zbmath.org/authors/?q=ai:ciraulo.francesco"Sambin, Giovanni"https://zbmath.org/authors/?q=ai:sambin.giovanniSummary: A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies, and we show that the former is a reflective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales.Uniformly based Cuntz semigroups and approximate intertwiningshttps://zbmath.org/1507.460392023-05-08T18:47:08.967005Z"Cantier, Laurent"https://zbmath.org/authors/?q=ai:cantier.laurentThis paper studies topological properties of abstract Cuntz semigroups. The main result develops an intertwining argument in the setting of inductive sequences of Cuntz semigroups. This is then applied to obtain classification results for *-homomorphisms from the C*-algebra of continuous functions on the unit circle to AF algebras.
Reviewer: Xin Li (Glasgow)Separated and prime compactificationshttps://zbmath.org/1507.540092023-05-08T18:47:08.967005Z"Razafindrakoto, A."https://zbmath.org/authors/?q=ai:razafindrakoto.andoThe author finds conditions under which certain compactifications of topological spaces can be obtained by composing the ultrafilter space monad with suitable reflectors. It is also shown that these compactifications inherit their categorical properties from the ultrafilter space monad. He observes that various constructions such as the prime open filter monad defined by \textit{H. Simmons} [Topology Appl. 13, 201--223 (1982; Zbl 0484.18005)], the prime closed filter compactification studied by \textit{H. L. Bentley} and \textit{H. Herrlich} [Quaest. Math. 25, No. 3, 381--396 (2002; Zbl 1036.54007)], as well as the separated completion monad studied by \textit{S. Salbany} [Commentat. Math. Univ. Carol. 23, 301--311 (1982; Zbl 0572.54008)] fall within the same categorical framework.
Reviewer: Minani Iragi (Johannesburg)Spatio-temporal domains: an overviewhttps://zbmath.org/1507.681752023-05-08T18:47:08.967005Z"Janin, David"https://zbmath.org/authors/?q=ai:janin.davidSummary: We consider the possibility of defining a general mathematical framework for the homogeneous modeling and analysis of heterogeneous spatio-temporal computations as they occur more and more in modern computerized systems of systems. It appears that certain fibrations of posets into posets, called here spatio-temporal domains, eventually provide a fully featured category that extends to space and time the category of cpos and continuous functions, aka Scott Domains, used in classical denotational semantics.
For the entire collection see [Zbl 1398.68027].The inf-translation for solving set minimization problemshttps://zbmath.org/1507.901882023-05-08T18:47:08.967005Z"Hamel, Andreas H."https://zbmath.org/authors/?q=ai:hamel.andreas-h"Heyde, Frank"https://zbmath.org/authors/?q=ai:heyde.frank"Visetti, Daniela"https://zbmath.org/authors/?q=ai:visetti.danielaIt is known that set-valued and vector-valued optimization problems may be recast as complete lattice-valued problems. The advantages of such a reformulation is the existence of a clear-cut solution concept and minimality. The problem then is to find a set which is simultaneously large enough to generate the infimum and small enough to include only minimizers. In this work, optimality conditions for such sets, based on the inf-translation, are presented in an abstract setting. The inf-translation reduces the solution set to a single point, which in turn, admits the application of more standard procedures. For functions with values in complete lattices of sets, scalarization results are proved for convex problems. Vector optimization problems are presented as examples.
Reviewer: K. C. Sivakumar (Chennai)