On Eilenberg–Moore algebras induced by chains

Let S be a fixed topological space. The contravariant Hom functor given by C(X) = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $Hom_Top(X, S)$\end{document} has an adjoint specified, on sets, by P(A)= SA and the composite, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $M = C \circ P$\end{document}, is a Monad on the category of sets. In this paper we characterize the category of Eilenberg–Moore Algebras associated with M in the special case where S is a linearly ordered space in its specialization order. The characterization is presented in terms of the notion of a dual frame which admits a C(S)-action.