DUALITY
The correspondences between algebraic and relational semantics, and the machinery that transports structure across them. Stone's theorem makes Boolean algebras and Stone spaces two views of one thing; Jónsson and Tarski extend it to Boolean algebras with operators and descriptive frames, which is why modal logic has both an algebraic and a Kripke semantics and why the canonical model construction works; Esakia does the same for Heyting algebras and intermediate logics.
The subject exists because this collection's two largest object-level divisions are dual. Algebraic individuates by the variety a logic is complete for; Modal individuates by the accessibility relation. Duality says those are the same information: the complex algebra of a frame and the ultrafilter frame of an algebra are inverse constructions, and a completeness proof on one side is a representation theorem on the other. Everything that looks like a coincidence between the two divisions—canonicity, correspondence, definability—is a duality result.
Entries here are the correspondences themselves. The theorems whose proofs they are—Sahlqvist correspondence, Goldblatt–Thomason definability—live where their statements live and are cross-listed.