README⤓ .txt 2026-10-22T165042.936 000000000000760 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.
Canonical Extensions⤓ .md 2026-10-22T085311.000 000000000039232 Implicit in Jónsson and Tarski (1951–52) as the complex algebra of the ultrafilter frame; isolated and generalized by Mai Gehrke and Bjarni Jónsson, "Bounded distributive lattices with operators" (1994) and "Bounded distributive lattice expansions" (2004). Gehrke, Nagahashi, and Venema (2005) gave the Sahlqvist theory algebraically. The point of the generalization: canonicity is not about Boolean algebras or about modal logic, and stating it without either shows what it is about.
Esakia Duality⤓ .md 2026-10-22T085315.000 000000000048272 Leo Esakia, "Topological Kripke models" (1974), which gave the duality for Heyting algebras; the Blok–Esakia theorem (Blok 1976, Esakia 1976) is its most consequential corollary. Esakia's Heyting Algebras: Duality Theory was published in English only in 2019, thirty-four years after the Russian original, which is part of why the subject was slow to travel.
Goldblatt-Thomason Theorem⤓ .md 2026-10-22T085021.000 000000000038992 Goldblatt and Thomason (1974). Characterizes modally definable frame classes. Frame conditions expressible by modal formulas. Closure under bounded morphic images, generated subframes and disjoint unions — plus reflection of ultrafilter extensions, which runs in the opposite direction from the other three.
Jonsson-Tarski Duality⤓ .md 2026-10-22T180154.561 000000000000880 Bjarni Jónsson and Alfred Tarski, "Boolean algebras with operators, Parts I and II" (American Journal of Mathematics, 1951–52) — written before Kripke semantics existed, and containing it. Goldblatt's "Metamathematics of modal logic" (1976) and Thomason's work made the connection explicit; Blackburn, de Rijke, and Venema's Modal Logic (2001) is where the field learned to teach it.
Priestley Duality⤓ .md 2026-10-22T180154.561 000000000000840 Hilary A. Priestley, "Representation of distributive lattices by means of ordered Stone spaces" (Bulletin of the London Mathematical Society 2, 1970), with the full categorical duality in "Ordered topological spaces and the representation of distributive lattices" (Proceedings of the London Mathematical Society 24, 1972). Marshall Stone had already represented bounded distributive lattices on their prime spectra in "Topological representations of distributive lattices and Brouwerian logics" (Časopis pro pěstování matematiky a fysiky 67, 1938), but his spaces are not Hausdorff; Priestley kept the same points — the prime filters — and added the inclusion order to a compact Hausdorff topology. William Cornish, "On H. Priestley's dual of the category of bounded distributive lattices" (Matematički Vesnik 12, 1975), proved the two presentations interchangeable. Davey and Priestley's Introduction to Lattices and Order (Cambridge, 1990; second edition 2002) is the standard textbook treatment.
Sahlqvist Correspondence⤓ .md 2026-10-22T090413.000 000000000034744 Henrik Sahlqvist (1975). Modal formulas with first-order correspondents. Syntactic class guaranteeing correspondence. Canonical frame validity. Automatic axiomatization.
Stone Duality⤓ .md 2026-08-20T065407.000 000000000037960 Marshall Stone, "The theory of representations for Boolean algebras" (1936) and "Applications of the theory of Boolean rings to general topology" (1937). Priestley extended it to bounded distributive lattices (1970). The theorem that made "algebra and topology are the same subject twice" a working method rather than a slogan.
CRITERIA⤓ .txt 2026-07-16T000353.000 000000000013168 Not sufficient: An algebraic semantics presented on its own (Algebraic, where the logic lives; Model-Theory for the study of algebraizability as such). A relational semantics presented on its own (Modal). A categorical formulation of a logic (Categorical)—a duality is an equivalence of categories, but the subject here is which two categories and what the equivalence preserves, not the category theory that states it.