Algebraic theories of compact pospaces
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).
scientific article; zbMATH DE number 1038769
scientific article