Representation theorems for directed completions of consistent algebraic L-domains

Abstract

In this paper, consistent algebraic L-domains are considered. One algebraic and two topological characterization theorems for their directed completions are given. It is proved that eliminating a set of maximal elements with empty interior from an algebraic L-domain results a consistent algebraic L-domain whose directed completion is just the given algebraic L-domain up to isomorphism. It is also proved that the category CALDOM of consistent algebraic L-domains and Scott continuous maps is Cartesian closed and has the category ALDOM of algebraic L-domains and Scott continuous maps as a full reflective subcategory.

When do abstract bases generate continuous lattices and L-domains

“…It is well known that the round ideal completion of an abstract basis is a domain (i.e., continuous dcpo) (cf., [1,3,5]). In addition, the ideal completion of a sup-semilattice with a least element (resp., cusl, L-cusl) is an algebraic lattice (resp., Scott domain, algebraic L-domain) (cf., [3,8,9]). Then it is natural to wonder when the round ideal completion of an abstract basis is a continuous lattice (resp., bc-domain, L-domain).…”

“…A poset with a least element is called a conditional upper semilattice with a least element (in short, cusl ) (see [3,9]) if every consistent finite subset has a supremum. A poset is called a locally conditional upper semilattice (in short, L-cusl ) (see [8]) if every principal ideal is a cusl. A sup-semilattice is a poset in which every nonempty finite subset has a supremum.…”