Lattices of Scott-closed sets
نویسندگان
چکیده
A dcpo P is continuous if and only if the lattice C(P ) of all Scottclosed 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 7→ C(P ) to a left adjoint functor from the category DCPO of dcpo’s to the category CPAlg of C-prealgebraic lattices.
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