For a number of locally finitely presentable categories K we describe the codensity monad full embedding all objects into K. We introduce concept D-ultrafilter on an object, where D is “nice” cogenerator prove that assigns to every object representing D-ultrafilters it. Our result covers e.g. sets, vector spaces, posets, semilattices, graphs and M-sets for finite commutative monoids M.

If the continuum hypothesis (CH) is true, then for any P point ultrafilter D (on the set of natural numbers) there exist initial segments of the Rudin-Keisler ordering, restricted to (isomorphism classes of) P points which lie above D, of order type N,. In particular, if D is an RK-minimal ultrafilter, then we have (CH) that there exist P-points with countably many constellations. 0. Introducti...

It is consistent (relative to ZFC) that each union of max{b, g} many families in the Baire space ω which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U , the cofinality of the reduced ultrapower ω/U is greater than max{b, g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained intr...

We show that if cov(M) = κ, where κ is a regular cardinal such that ∀λ < κ(2 ≤ κ), then for every unbounded directed family H of size κ there is an ultrafilter UH such that the relativized Mathias forcing M(UH) preserves the unboundedness of H. This improves a result of M. Canjar (see [4, Theorem 10]). We discuss two instances of generic ultrafilters for which the relativized Mathias forcing pr...

Filtration by means of a Diaflo ultrafilter was used to concentrate three 1,000-ml lots and one 3,000-ml lot of tissue culture-grown Rift Valley fever virus. Quantitation of both infectivity and total protein was achieved. Water treatment with continued ultrafiltration of the virus concentrate provided a final virus product approximately 99.25% free of low-molecular-weight materials originally ...

The ultrapower T * of an arbitrary ordered set is introduced as infinitesimal extension . It obtained the equivalence classes sequences in , where corresponding relation generated by a free ultrafilter on natural numbers. established that always satisfies Cantor’s property, while one can give necessary and sufficient conditions for so would be complete or it fulfill open completeness respective...

We show that if cov(M) = κ, where κ is a regular cardinal such that ∀λ < κ(2 ≤ κ), then for every unbounded directed family H of size κ there is an ultrafilter UH such that the relativized Mathias forcing M(UH) preserves the unboundedness of H. This improves a result of M. Canjar (see [4, Theorem 10]). We discuss two instances of generic ultrafilters for which the relativized Mathias forcing pr...

We show that if (S,+) is a commutative semigroup which can be embedded in the circle group T, in particular if S = (N,+), then all nonprincipal, strongly summable ultrafilters on S are sparse and can be written as sums in βS only trivially. We develop a simple condition on a strongly summable ultrafilter which guarantees that it is sparse and show that this holds for many ultrafilters on semigr...

We study the I-ultrafilters on ω, where I is a a collection of subsets of a set X , usually R or ω1. The I-ultrafilters usually contain the P -points, often as a small proper subset. We study relations between I-ultrafilters for various I, and closure of I-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I-ultrafilters always exist.

Let L be a finite first-order language and 〈Mn | n < ω〉 be a sequence of finite L-models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space 2 is non-empty, then there is a non-principal ultrafilter U over ω such that the corresponding ultraproduct Q U Mn is infinite and has an automorphism that is not ind...