Related papers: Divisibility orders in $\beta N$
After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-Cech compactification \beta N, we establish some results about one of such extensions. This provides us with one possible divisibility…
We continue the research of an extension $\widetilde{\mid}$ of the divisibility relation to the Stone-\v Cech compactification $\beta N$. First we prove that ultrafilters we call prime actually possess the algebraic property of primality.…
The paper first covers several properties of the extension of the divisibility relation to a set ${}^*\hspace{-0.5mm}N$ of nonstandard integers. After that, a connection is established with the divisibility in the Stone-\v{C}ech…
An extension of the divisibility relation on $\mathbb{N}$ to the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so…
In this paper we will prove that all the elements in the smallest ideal K($\beta$N) in the semigroup of the Stone Cech compactification ($\beta$N,.) of the discrete semigroup of natural numbers N under multiplication constitute a single…
In this work we analyze some topological properties of the remainder $\partial M:=\beta_s^* M\setminus M$ of the semialgebraic Stone-C\v{e}ch compactification $\beta_s^* M$ of a semialgebraic set $M\subset{\mathbb R}^m$ in order to…
To work more accurately with elements of the semigroup of the Stone Cech compactification of the discrete semigroup of natural numbers N under multiplication. We divided these elements into ultrafilters which are on finite levels and…
We study Medvedev reducibility in the context of set theory -- specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li \cite{HaLi}, we show that the Medvedev degrees of countable ordinals are far from…
We further investigate a divisibility relation on the set $\beta N$ of ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every…
In tri-partite systems, there are three basic biseparability, $A$-$BC$, $B$-$CA$ and $C$-$AB$ biseparability according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the…
The Stone-Cech compactification of the natural numbers bN, or equivalently, the space of ultrafilters on the subsets of omega, is a well-studied space with interesting properties. If one replaces the subsets of omega by partitions of omega,…
In [1] the authors showed some basic properties of a pre-order that arose in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and they presented its generalization to ultrafilters, which is…
We survey some recent results about the order structure of various kinds of ultrafilters. More precisely, we study Rudin-Keisler and Tukey reducibility in classes of selective, stable ordered-union, and P-point ultrafilters. Although these…
The following is an open problem in topology: Determine whether the Stone-\v{C}ech compactification of a widely-connected space is necessarily an indecomposable continuum. Herein we describe properties of $X$ that are necessary and…
We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-\v{C}ech compactification $\beta…
We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations $\mid_M$ and…
We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega$ implies that the modal logic S4.1.2 is complete with respect to the \v{C}ech-Stone compactification of the natural numbers,…
Let $c:\mathbb Z^2\to \{0, 1\}$ be a configuration with a non-trivial annihilator. We show that if $c$ is weakly periodic then the directions of periodicity in a minimal weakly periodic decomposition of $c$ can be detected from the…
Let $(W,S)$ be an arbitrary Coxeter system. For each word $\omega$ in the generators we define a partial order--called the {\sf $\omega$-sorting order}--on the set of group elements $W_\omega\subseteq W$ that occur as subwords of $\omega$.…
Meshalkin's theorem states that a class of ordered p-partitions of an n-set has at most $\max \binom{n}{a_1,...,a_p}$ members if for each k the k'th parts form an antichain. We give a new proof of this and the corresponding LYM inequality…