Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (innuenced by and abstracting over both Goldblatt's 17] and Urquhart's 46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality. In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the JJ onsson and Tarski representation results 30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing , Dunn's project of gaggle theory 13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the JJ onsson-Tarski additive operators. Representation of`-algebras is extended to full duality. In part III we discuss applications in logic of the framework developed. Specii-cally, logics with restricted structural rules give rise to lattices with normal operators (in our sense), such as the Full Lambek algebras (FL-algebras) studied by Ono in 36]. Our Stone-type representation results can be then used to obtain canonical constructions of Kripke frames for such systems, and to prove a duality of algebraic and Kripke semantics for such logics. version appeared as a technical report at I.U. with title \Lattices with Additional Operators". Parts or preliminary versions have been presented at the universities of Toronto, Pittsburgh, Amsterdam, at the annual 1994 ASL meeting in Gainsville, Florida, and of course at Indiana and Leicester. Many thanks to the participants of the respective seminars for useful comments and suggestions.