By Yde Venema
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Extra resources for Algebras and Coalgebras
12) the carrier L of L forms a dense subset of the σ-topology over the carrier Lσ . Also observe that f σ and f π are extensions of f because all elements of L are isolated points of f , and that for monotone f , (37) agrees with (32). Finally, it is not just the definitions that translate to the more general setting of lattice expansions (that is, lattices with additional operations), the same holds for the theory. 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.
Un 1 A σ (t ) , as desired. ✷ As a sample application, we show how Sahlqvist canonicity is an easy consequence of the previous theorem. 21 Sahlqvist equations are canonical over the class of all Boolean algebras with τ -operators. Proof. First we treat inequalities of the form ϕ(β1 , . . , βn ) ≤ ψ, where ϕ only uses ∧, ∨ and modalities, all βi are boxed atoms, and ψ is positive. But then it is immediate by the previous proposition that ϕ(β1 , . . , βn ) is stable, while ψ is expanding. 17.
Before we define the concept of density, we introduce some preliminary notions. Given a completion C of the Boolean algebra B, we call an element c ∈ C closed (open) if c is the meet (join, respectively) in C of elements in B. We let KC (B) and OC (B) denote the collections of closed and open elements, respectively. Objects (such as the elements of B) that are both closed and open are called clopen. 21. In the sequel, we may write KC , K(B), or even K, instead of KC (B), if the suppressed details are clear from context; and similarly for the set OC (B).