Algebraic theories of compact pospaces

Let CmptPoSp denote the category of compact pospaces with continuous monotone maps and let PoSet denote the category of partially ordered sets and monotone maps. In this paper we show that the forgetful functor G: CmptPoSp → PoSet is monadic; that is, G has a left-adjoint and CmptPoSp is isomorphic to the category of algebras PoSetB for the monad B on PoSet induced by the adjunction. This result, which is an asymmetric version of Manesʹ theorem, shows that the notion of compact pospace is algebraic in a precise sense and provides a useful tool for investigating the category CmptPoSp. As a corollary we obtain the theorem of Simmons and Wyler which says that CmptPoSp is also algebraic over the category of topological spaces and continuous maps. This makes explicit the connection between the Salbany and the prime Wallman compactifications. We also give an explicit construction—as the prime spectrum of the lattice of upper sets—of the Stone-Čech-Nachbin order compactification for a discrete ordered space.