Algebraic theories of compact pospaces
type: Item
instance of: scholarly article
title: Algebraic theories of compact pospaces
zbMATH Open document ID: 0877.18003
DOI: 10.1016/S0166-8641(96)00117-4
author: Bob Flagg
published in: Topology and its Applications
publication date: 1997-11-06T00:00:00Z
review text: If \({\mathcal C}mpt{\mathcal P}o{\mathcal S}p\) denotes the category of compact pospaces (compact topological spaces \((X,\tau)\) together with a partial ordering \(\leq \) of \(X\) which is closed in the product space \((X,\tau)\times (X,\tau)\)) with continuous monotone maps and \({\mathcal P}o{\mathcal S}et\) denotes the category of partially ordered sets and monotone maps, the author proves that the forgetful functor \(G:{\mathcal C}mpt{\mathcal P}o{\mathcal S}p\rightarrow{\mathcal P}o{\mathcal S}et\) is monadic (i.e., has a left-adjoint and \({\mathcal C}mpt{\mathcal P}o{\mathcal S}p\) is isomorphic to the category of algebras \({\mathcal P}o{\mathcal S}et^B\), for a monad \(B\) induced by the adjunction). This is an asymmetric version of Manes' criterion which shows that in some sense the notion of compact pospace is algebraic. Reviewer's remarks: The theorem of Simmons and Wyler \(({\mathcal C}mpt{\mathcal P}o{\mathcal S}p\) is algebraic over \({\mathcal T}op)\) is obtained as a corollary. This makes explicit the connection between the Salbany and the prime Wallman compactifications. Finally, also an explicit construction of the Stone-Čech-Nachbin order compactification for discrete ordered spaces is given (as the prime spectrum of the lattice of upper sets).
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D25
zbMATH DE Number: 1038769
zbMATH Keywords: monad
zbMATH Keywords: compact pospace
zbMATH Keywords: algebra
zbMATH Keywords: compactification
zbMATH Keywords: prime filter
zbMATH Keywords: prime spectrum
zbMATH Keywords: asymmetric topology
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1038769