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**11 - 12**of**12**### Pseudo-Retract Functors for Local Lattices and Bifinite L-Domains

"... Recently, a new category of domains used for the mathematical foundations of denotational semantics, that of L-domains, has been under study. In this paper we consider a related category of posets, that of local lattices. First, a completion operator taking posets to local lattices is developed, ..."

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Recently, a new category of domains used for the mathematical foundations of denotational semantics, that of L-domains, has been under study. In this paper we consider a related category of posets, that of local lattices. First, a completion operator taking posets to local lattices is developed, and then this operator is extended to a functor from posets with embedding-projection pairs to local lattices with embedding-projection pairs. The result of applying this functor to a local lattice yields a local lattice isomorphic to the rst; this functor is a pseudo-retract. Using the functor into local lattices, a continuous pseudo-retraction functor from ω-bifinite posets to ω-bifinite L-domains can be constructed. Such a functor takes a universal domain for the ω-bifinite posets to a universal domain for the ω-bifinite L-domains. Moreover, the existence of such a functor implies that, from the existence of a saturated universal domain for the ω-algebraic bifinites, we can conclude...

### A Logical Approach to Stable Domains

, 2006

"... Building on earlier work by Guo-Qiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic L-domains. Disjunctions in the ..."

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Building on earlier work by Guo-Qiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic L-domains. Disjunctions in the logic can be indexed by arbitrary sets (as in geometric logic) but must be provably disjoint. This raises several technical issues which have to be addressed before clean notions of axiom system and theory can be defined. We show soundness and completeness of the proof system with respect to distributive disjunctive semilattices, and prove that every such semilattice arises as the Lindenbaum algebra of a disjunctive theory. Via stable Stone duality, we show how to use disjunctive propositional logic for a logical description of algebraic L-domains.