# representation of L-domains

Date: Mon, 26 Aug 91 12:00:23 -0400

Disjunctive Systems and L-Domains

Guo-Qiang Zhang
AI Lab
Department of EECS
The University of Michigan
Ann Arbor,  MI 48109
e-mail: gqz@caen.engin.umich.edu
(full manuscript available on request)

Disjunctive systems are a representation of L-domains. They use sequents
of the form $X\vdash Y$, with $X$ finite and $Y$ pairwise disjoint.  We
show that for any disjunctive system, its elements ordered by inclusion
form an L-domain. On the other hand, via the notion of stable
neighborhoods, every L-domain can be represented as a disjunctive system.
These results are then put in a more general setting of equivalence of
categories.

A natural classification of domains is obtained in terms of the style of
the entailment: when $\elements{ X} =2$ and $\elements{Y} =0$ disjunctive
systems determine coherent spaces; when $\elements{Y} \leq 1$ they
represent Scott domains; when either $\elements{X} =1$ or $\elements{Y}=0$
the associated cpos are distributive Scott domains; and, finially, without
any restriction, disjunctive systems give rise to L-domains.