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By Yde Venema

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16 Although the Sahlqvist canonicity theorem takes care of most of the canonical formulas that one encounters in practice, it certainly does not cover the concept completely. For instance, G ORANKO & VAKARELOV [49] widen the class to that of so-called inductive formulas, see Chapter ?? of HBML for some ´ discussion. J ONSSON [69] generalizes an example of F INE [25] to the result that for every positive formula ϕ(x), the equation ϕ(x ∨ y) ≈ ϕ(x) ∨ ϕ(y) is canonical. And of course, there are individual examples of canonical formulas, such as the conjunction of the transitivity axiom 4 and the McKinsey axiom ✷✸x ≤ ✸✷x, cf.

To mention just one example: one may prove that any equation s ≈ t is canonical provided that all the primitive symbols (including the join operation ∧) occurring in s and t are interpreted as operators. Details can be found in for instance G EHRKE & H ARDING [28]. The second generalization that we want to mention involves other ways of completing lattices and lattice expansions, such as the MacNeille completion, which generalizes Dedekind’s construction of the reals from the rationals to arbitrary partial orders.

Proof. Recall from the topological duality that B∗ = Uf B, B is a differentiated and compact field of sets. By the comments made above it should be clear that P(Uf B) is a canonical extension of B. For unicity, suppose that C is a canonical extension of B. We leave it as an exercise for the reader to verify that, by compactness, the map F → F forms a dual (that is, order-reversing) isomorphism between the lattice Fi (B), ⊆ and the induced ordering on the set K(B) of closed elements. Its inverse is given by the map p → {a ∈ B | a ≥ p}.

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