Alexandrov compactification of relational algebras
type: Item
instance of: scholarly article
title: Alexandrov compactification of relational algebras
zbMATH Open document ID: 0532.18003
DOI: 10.1007/BF01192820
author: Axel Möbus
published in: Archiv der Mathematik
publication date: 1983-01-01T00:00:00Z
review text: It is well-known that compact spaces can be viewed as algebras over the ultrafilter monad where each algebra structure is given by a function which sends every ultrafilter to its unique limit point. Arbitrary topological spaces cannot be obtained by this method. They can, however, be viewed as so-called relational algebras over the ultrafilter monad where each algebra structure is given by a correspondence which pairs certain ultrafilters with their limit points. In this setting, it is also possible to characterize locally compact spaces. The author investigates conditions under which the one-point (or Alexandrov) compactification of locally compact spaces can be carried over to locally compact relational algebras for other monads. The general setting is a base category \({\mathcal C}\) with a calculus of relations (or correspondences) and representability for partial morphisms. In such a representation \(t_ B: B\to \tilde B,\tilde B\) serves as a generalization of the usual \(B\cup \{\infty \}\). One-element completion of partial algebras and of ordered sets are the only examples cited beside the one-point compactification of topological spaces.
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 08A55
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 3843974
zbMATH Keywords: relational algebras
zbMATH Keywords: compactification
zbMATH Keywords: completion
zbMATH Keywords: monad algebras
zbMATH Keywords: algebras over ultrafilter monad
zbMATH Keywords: compact spaces
reviewed by: J. Sonner
MaRDI profile type: Publication
cites work: On extending congruences from partial algebras
cites work: Q5664055
cites work: The Spectral Theory of Distributive Continuous Lattices
cites work: Relations in categories
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cites work: On a method of construction of abstract algebras
cites work: Korrespondenzen in abelschen Kategorien
cites work: Q5666704
full work available at URL: https://doi.org/10.1007/bf01192820
OpenAlex ID: W2092057407
description: scientific article; zbMATH DE number 3843974