By Yde Venema
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Earlier on we defined Aσ via a concrete construction, namely, as the ‘double dual’ (A• )+ : the complex algebra of the ultrafilter frame of A. In this section we will take a rather more abstract approach in which we first consider the canonical extension Bσ of the Boolean reduct B of A; this Bσ is not constructed but axiomatically characterized as the (modulo isomorphism) unique completion of B in which B is dense and compact. Then the property of density suggests a canonical way to extend the interpretation of the operators on B to operations on Bσ , thus providing the canonical extension Aσ of A.
For instance, G ORANKO & VAKARELOV  widen the class to that of so-called inductive formulas, see Chapter ?? of HBML for some ´ discussion. J ONSSON  generalizes an example of F INE  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.  for an algebraic proof. As we mentioned, a second way to arrive at canonical varieties of BAOs proceeds via a model-theoretic road.
Tn ) A ≤ (∇A )σ ◦ tA 1 , . . , tn σ σ σ A σ = ∇A ◦ (tA 1 ) , . . , (tn ) σ σ A ≤ ∇ A ◦ tA 1 , . . , tn σ σ = tA . 18, and the fourth step is by the inductive σ hypothesis and the monotonicity of ∇A . 44 σ For part (ii) and (iii) it suffices to prove that tA ≤ (tA )σ , since the opposite inequality holds by part (i). In the case of part (ii) this follows from a straightforward induction, whereas for part (iii) we need the principle of matching topologies. σ σ σ Aσ = (sA )σ ◦ (uA )σ , . . , (uA )σ Let t be as described in part (iii), then tA = sA ◦ uA n 1 , .