Embedding partially ordered spaces in topological semilattices

Abstract:

A partial order $\Gamma$ on a compact space S is called continuous if $\Gamma$ is a closed subset of $S \times S$. In this paper, we define and study an embedding $\Phi$ of the arbitrary compact continuously partially ordered space $(S,\Gamma )$ into a corresponding compact topological semilattice ${S_\Gamma }$. We show that the structure of ${S_\Gamma }$ entirely determines the structure of $(S,\Gamma )$. We prove that the inverse images under $\Phi$ of components in ${S_\Gamma }$ are the order components of $(S,\Gamma )$, where elements a and b of S are defined to be in the same order component of $(S,\Gamma )$ if there exists no continuous monotonic map $f:(S,\Gamma ) \to \{ 0,1\}$ which separates a and b. Finally, we show that ${S_\Gamma }$ is connected if and only if $(S,\Gamma )$ has only one order component.