Canonical Extensions

Origin. 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.

Models. Take an algebra and complete it, in the way the ultrafilter construction completes a Boolean algebra — but state the completion by a universal property rather than by ultrafilters, and it applies to bounded distributive lattices, residuated lattices, and lattice expansions generally. Canonicity is then the question of which equations survive the completion, and the answer is a purely algebraic one that happens to explain modal correspondence theory.

Formalism.

The canonical extension A^σ of A: A completion satisfying two conditions. Density: every element of A^σ is a join of meets of elements of A. Compactness: if ⋀X ≤ ⋁Y for X, Y ⊆ A, then ⋀X′ ≤ ⋁Y′ for finite X′ ⊆ X, Y′ ⊆ Y. These determine A^σ up to isomorphism fixing A — the universal property replaces the construction.

Recovering Jónsson–Tarski: For a Boolean algebra B, B^σ ≅ ℘(X_B) — the powerset of the Stone space. For a BAO, B^σ = (B₊)⁺, the complex algebra of the ultrafilter frame. So the 1951 construction is the σ-extension, and the definition above says what it was.

Extending operations — two ways: σ-extension (lower): f^σ(u) = ⋁{⋀ f[x] : x ∈ A, x ≥ ...} π-extension (upper): the dual. They agree on operators that are (co)continuous; where they differ, the choice matters. Additive operations: f^σ = f^π. This is why modal ◇ is well-behaved and general operations are not.

Canonicity: An equation s = t is canonical if A ⊨ s = t implies A^σ ⊨ s = t. A canonical variety is closed under canonical extension. Canonical axioms give complete logics — the algebraic statement of the completeness-via-canonical-models argument.

Sahlqvist, algebraically (Gehrke–Nagahashi–Venema): Sahlqvist inequalities are exactly those built so that the σ- and π-extensions can be shown to agree on the relevant terms. Canonicity follows from that agreement; correspondence follows from the density condition. The syntactic Sahlqvist shape is a sufficient condition for a semantic property of the extension.

Beyond Boolean: Bounded distributive lattices with operators: same theory, Priestley duality underneath. Residuated lattices and substructural logics: canonicity results are harder and partial. The generalization is what made duality available to Structural and not only to Modal.

Symbols.

Metatheory. Stating canonical extension by density and compactness rather than by ultrafilters is what freed the theory: the ultrafilter construction needs a Boolean algebra, and the universal property needs only a lattice, so the whole Sahlqvist apparatus transfers to distributive and residuated settings where no ultrafilter frame exists. That the σ- and π-extensions coincide for additive operations and diverge otherwise is the technical heart, and it explains a fact modal logicians knew without knowing why: normal modal logic is well-behaved because ◇ is additive, and non-normal logics are not because their operators are not. Fine's incompleteness examples are algebraically non-canonical varieties.

Applies to. Canonicity and completeness for modal, distributive-modal, and substructural logics. Sahlqvist theory in its algebraic form. Duality for lattices with operations, where the extension is the object the duality is stated over. Coalgebraic logic and the generalization of correspondence beyond Kripke frames.

Limitations. The extension is enormous and non-constructive: A^σ is built from ultrafilters or from an equivalent choice principle, and nothing about it is effective. Canonicity is sufficient for completeness and not necessary, so the theory explains the well-behaved cases and is silent about the rest — and Fine's and Thomason's incompleteness results sit in that silence. For residuated lattices the canonicity results are partial: the non-integral, non-commutative cases have no general theorem, which is why Structural benefits from the framework less than Modal does.

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