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Lattices of Scott-closed sets

Weng Kin Ho; Dong Sheng Zhao — 2009

Commentationes Mathematicae Universitatis Carolinae

A dcpo $P$ is continuous if and only if the lattice $C (P)$ of all Scott-closed subsets of $P$ is completely distributive. However, in the case where $P$ is a non-continuous dcpo, little is known about the order structure of $C (P)$ . In this paper, we study the order-theoretic properties of $C (P)$ for general dcpo’s $P$ . The main results are: (i) every $C (P)$ is C-continuous; (ii) a complete lattice $L$ is isomorphic to $C (P)$ for a complete semilattice $P$ if and only if $L$ is weak-stably C-algebraic; (iii) for any two complete semilattices...

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