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A survey on divisibility of ultrafilters

Boris Šobot

Abstract

An extension of the divisibility relation on $\mathbb{N}$ to the set $β\mathbb{N}$ 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 far, adding several results connecting the themes of different stages of the research. The highlights include: separation of $β\mathbb{N}$ into the lower part $L$ (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter with given basic ingredients; existence and number of successors and predecessors of a given divisibility class; extending the congruence relation (in two ways) and checking properties of the obtained relations.

A survey on divisibility of ultrafilters

Abstract

An extension of the divisibility relation on to the set of ultrafilters on was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so far, adding several results connecting the themes of different stages of the research. The highlights include: separation of into the lower part (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter with given basic ingredients; existence and number of successors and predecessors of a given divisibility class; extending the congruence relation (in two ways) and checking properties of the obtained relations.
Paper Structure (9 sections, 34 theorems, 22 equations)

This paper contains 9 sections, 34 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and notation
  3. The nonstandard method
  4. The lower part $L$
  5. Levels, patterns and $=_\sim$-classes in $L$
  6. Generalized patterns
  7. Congruence modulo an ultrafilter
  8. Other divisibility-like relations
  9. Remarks and open problems

Key Result

Theorem 1.1

For every coloring $f:\mathbb{N}\rightarrow\{1,2,\dots,k\}$ there is an infinite $A\subseteq\mathbb{N}$ such that the set $FS(A):=\{a_1+a_2+\dots+a_n:n\in\mathbb{N}\land a_1,a_2,\dots,a_n\in A\hbox{are distinct}\}$ is monochromatic.

Theorems & Definitions (62)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4: So5, Lemma 4.1.
  • Example 3.1
  • Proposition 3.2
  • Definition 3.3
  • Theorem 3.4: DN2, Theorem 11.5.12
  • Theorem 3.5: P, Theorem 3.4
  • ...and 52 more