We study a class of operators on Turing degrees arising naturally from ultrafilters. Suppose U is a nonprincipal ultrafilter on ω. We can then view a sequence of sets A = (Ai)i∈ω as an “approximation” of a set B produced by taking the limits of the Ai via U : we set limU (A) = {x : {i : x ∈ Ai} ∈ U}. This can be extended to the Turing degrees, by defining δU (a) = {limU (A) : A = (Ai)i∈ω ∈ a}. ...

We prove the following: Theorem A. If D is a (λ+, κ)-regular ultrafilter, then either (a) D is (λ, κ)-regular, or (b) the cofinality of the linear order ∏ D〈λ, <〉 is cf κ, and D is (λ, κ′)-regular for all κ′ < κ. Corollary B. Suppose that κ is singular, κ > λ and either λ is regular, or cf κ < cf λ. Then every (λ+n, κ)-regular ultrafilter is (λ, κ)-regular. We also discuss some consequences and...

We consider selective ultrafilters on the collection FIN of all finite nonempty subsets of N. If countablesupport side-by-side Sacks forcing is applied, then every selective ultrafilter in the ground model generates a selective ultrafilter in the extension. We also show that selective ultrafilters localizes the Parametrized Milliken Theorem, and that selective ultrafilters are Ramsey.

This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality. It is wellknown that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence. Stone-coalgebras (coalgebras over the category of Stone spaces), on the ot...