Related papers: Laver ultrafilters
Let $\lambda$ and $\kappa$ be cardinal numbers such that $\kappa$ is infinite and either $2\leq \lambda\leq \kappa$, or $\lambda=2^\kappa$. We prove that there exists a lattice $L$ with exactly $\lambda$ many congruences, $2^\kappa$ many…
There exist two known canonical types of ultrafilter extensions of first-order models; one comes from modal logic and universal algebra, another one from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups as…
We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. Using that relation we prove (ZFC) that the class of strict $J_{\omega^\omega}$-ultrafilters, introduced by J. E. Baumgartner in…
We discuss the connection between various orders on the class of all the ultrafilters and certain compactness properties of abstract logics and of topological spaces. We present a model theoretical characterization of Comfort order. We…
We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural…
We show that if $\lambda^{<\kappa} = \lambda$ and every normal filter on $P_\kappa\lambda$ can be extended to a $\kappa$-complete ultrafilter then so does every $\kappa$-complete filter on $\lambda$. This answers a question of Gitik.
We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $ \lambda $ is a singular cardinal, $ \lambda ' < \lambda $, and the ultrafilter $D$ is $ \kappa $-decomposable for all…
The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole…
Tensor products of ultrafilters have special combinatorial features closely related to Ramsey's Theorem, making them useful tools in applications. Here we first review their fundamental properties and isolate some new ones, including a…
It was recently shown that arbitrary first-order models canonically extend to models (of the same language) consisting of ultrafilters. The main precursor of this construction was the extension of semigroups to semigroups of ultrafilters, a…
Assuming the Generalized Continuum Hypothesis, this paper answers the question: when is the tensor product of two ultrafilters equal to their Cartesian product? It is necessary and sufficient that their Cartesian product is an ultrafilter;…
We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal $\lambda$ for which there is $\mu < \lambda \leq 2^\mu$, we construct…
We overview classifications of simple infinite-dimensional complex $\mathbb{Z}$-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose…
In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set…
We improve Galvin's Theorem for ultrafilters which are p-point limits of p-points. This implies that in all the canonical inner models up to a superstrong cardinal, every $\kappa$-complete ultrafilter over a measurable cardinal $\kappa$…
We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.
A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative…
We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the…
We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as…
Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable'…