Jónsson–Tarski Duality

Origin. 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.

Models. Stone duality plus one operator. A Boolean algebra with a normal additive operator is dual to a Stone space with a relation — which is to say, to a Kripke frame carrying a topology. The relation is recovered from the algebra as inclusion among ultrafilters modulo the operator; the algebra is recovered from the frame as its complex algebra. Modal logic's two semantics are one construction read in two directions, and Jónsson and Tarski had the theorem fifteen years before anyone read it that way.

Formalism.

Boolean algebra with operators (BAO): (B, ∧, ∨, ¬, 0, 1, f) with f normal (f(0) = 0) and additive (f(a ∨ b) = f(a) ∨ f(b)). The modal ◇ is exactly such an f.

Complex algebra of a frame: For a frame F = (W, R), take B = ℘(W) with ◇X = {w : ∃v (wRv and v ∈ X)} This is F⁺, the complex algebra. Additivity is automatic; normality is ◇∅ = ∅.

Ultrafilter frame of an algebra: For a BAO B, take W = ultrafilters, and U R V iff {◇a : a ∈ V} ⊆ U This is B₊, the ultrafilter (canonical) frame.

The canonical embedding: B ↪ (B₊)⁺ Every BAO embeds in the complex algebra of its ultrafilter frame — the Jónsson–Tarski theorem. The image is not all of it: (B₊)⁺ is the canonical extension, and it is strictly larger.

Descriptive general frames: Frames alone are not dual to algebras: ℘(W) is too big. A general frame (W, R, A) carries an admissible set algebra A ⊆ ℘(W). Descriptive = differentiated, tight, and compact. Then: BAO ≃ DGF^op, an equivalence.

Why canonical models work: The canonical model of a normal modal logic is the ultrafilter frame of its Lindenbaum algebra. Completeness proofs are the Jónsson–Tarski construction, run without knowing it.

Canonicity: A formula is canonical if it is preserved from B to (B₊)⁺. Canonical formulas axiomatize complete logics. Sahlqvist formulas are canonical — that is the algebraic content of Sahlqvist's theorem.

Symbols.

Metatheory. That Jónsson and Tarski had relational semantics for modal logic in 1951 and no one noticed until the 1970s is the field's standing embarrassment and its clearest evidence for what duality buys: the same theorem, unread on the algebraic side, was rediscovered as Kripke semantics and became the subject. The general-frame refinement is what makes the correspondence exact — plain Kripke frames are dual to nothing, because complex algebras are complete and atomic while arbitrary BAOs are neither, and the admissible sets are precisely the repair. Incompleteness results (Thomason, Fine) are then algebraic facts: a logic incomplete for frames is complete for general frames, always, because the algebraic semantics never fails.

Applies to. Every completeness proof in modal logic. Correspondence theory and canonicity. Coalgebraic logic, which generalizes the construction past Boolean algebras. Description logics and dynamic logics, whose semantics are BAOs with several operators. The relation between Algebraic and Modal in this collection, which this theorem is.

Limitations. The canonical extension is not constructive and not small: (B₊)⁺ is vastly larger than B, and canonicity is a fragile property — most modal formulas are not canonical, and no syntactic characterization of the canonical ones exists beyond Sahlqvist's sufficient condition. The duality needs descriptive general frames, which are technically necessary and philosophically awkward: a Kripke frame is an intuitive object and a general frame is a Kripke frame with bookkeeping. Non-normal modal logics fall outside entirely and need neighbourhood duality instead.

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