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By Yde Venema

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19 Let A be a τ -expanded Boolean algebra, and t a τ -term. Then σ (i) If A interprets all connectives in t as operators, then tA is (γ ↑ , γ ↑ )-continuous. σ (ii) If A interprets all connectives in t as additive maps, then tA is (σ ↓ , σ ↓ )-continuous. σ (iii) If A interprets all connectives in t as multiplicative maps, then tA is (σ ↑ , σ ↑ )-continuous. Proof. 18 for the induction step. For the induction base, note that the projection maps are both join- and meet preserving, and hence, their canonical extensions have all the continuity properties mentioned in the statements of this proposition.

39, and closed under taking ultrafilter extensions as a corollary of Theorem ?? in HBML. 19 then yields the desired result. 3 Interpolation In the last part of this section we discuss another fundamental property of logics: interpolation. Interpolation is important for applications because it allows reasoning systems to be set up in a modular way. Since we have confined our attention to logics in the form of sets of theorems, the version of interpolation that we will consider here is the following.

4 Composite maps We now investigate the interaction between composing maps between Boolean algebras and taking their canonical extensions. That is, we will take a look at the relation between the maps (gf )σ and g σ f σ for maps f : A → A , and g : A → A . We are obviously eager to find cases in which we have (gf )σ = g σ f σ , but also conditions under which one of the inequalities (≤ or ≥) apply will turn out to be of interest. As we will see shortly, many of these conditions can naturally be described in topological terms.

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Algebras and Coalgebras by Yde Venema

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