Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics

Part I of this paper is developed in the tradition of Stone-type dualitiesrΓ where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46])Δdentifying 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 IIΓwe consider lattice-ordered algebras (lattices with additional operators)Γ extending the Jónsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developingΓ and indeed completingΓDunn's project of gaggle theory [13Γ14]. We consider general lattices (rather than Boolean algebras)Γwith a broad class of operatorsΓwhich we dubb normalΓand which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. In part III we discuss applications in logic of the framework developed. SpecificallyΓ logics with restricted structural rules give rise to lattices with normal operators (in our sense)Γsuch as the Full Lambek algebras (FL-algebras) studied by Ono in [36]. Our Stonetype representation results can be then used to obtain canonical constructions of Kripke frames for such systemsΓand to prove a duality of algebraic and Kripke semantics for such logics. © 1997 Kluwer Academic Publishers.