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A Wallace semigroup whose every finite power is countably compact

Juan Luis Jaisuño Fuentes-Maguiña, Vinicius de Oliveira Rodrigues, Artur Hideyuki Tomita

Abstract

We show that, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, there exists a Wallace semigroup whose infinite countable power is the least power which fails to be countably compact. This answers positively Question 9.4 of \cite{Tomita15}.

A Wallace semigroup whose every finite power is countably compact

Abstract

We show that, assuming the existence of incomparable selective ultrafilters, there exists a Wallace semigroup whose infinite countable power is the least power which fails to be countably compact. This answers positively Question 9.4 of \cite{Tomita15}.
Paper Structure (8 sections, 6 theorems, 2 equations)

This paper contains 8 sections, 6 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. A bit of history
  3. Basic results and notation
  4. A briefing on stacks
  5. Main result
  6. $\mathbb H$ has no semigroup topology which makes its $\omega$-th power countably compact
  7. Comments and questions
  8. Acknowledgements

Key Result

Lemma 2.3

Let $p$ be a selective ultrafilter, $m \in \omega$ and $(h_i: i <m)$ be sequences in $\mathbb Z^{(\mathfrak c)}$ which are $p$-independent mod constants (with respect to $G=\mathbb Z^{(\mathfrak c)}$). Then there exists $A\in p$, a positive integer $N$ and a integer stack $\mathop{\mathrm{\mathcal{S

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Lemma 5.4., kp
  • Remark 2.4
  • Lemma 2.5: Lemma 8.4, Tomita15
  • Definition 2.6
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 6 more