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Keywords:
domain; complete semilattice; Scott-closed set; C-continuous lattice; C-algebraic lattice
domain; complete semilattice; Scott-closed set; C-continuous lattice; C-algebraic lattice
Summary:
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 $P$ and $Q$, $P$ and $Q$ are isomorphic if and only if $C(P)$ and $C(Q)$ are isomorphic. In addition, we extend the function $P\mapsto C(P)$ to a left adjoint functor from the category {\bf DCPO} of dcpo's to the category {\bf CPAlg} of C-prealgebraic lattices.
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