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Atomicity of Boolean algebras and vector lattices in terms of order convergence

Antonio Avilés, Eugene Bilokopytov, Vladimir G. Troitsky

Abstract

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.

Atomicity of Boolean algebras and vector lattices in terms of order convergence

Abstract

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.
Paper Structure (4 sections, 18 theorems)

This paper contains 4 sections, 18 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The case of a Boolean algebra
  4. The case of a vector lattice

Key Result

Lemma 2.1

For any proper ideal $R$ in a Boolean algebra $P$ there is an ultrafilter $Q\subseteq P\backslash R$.

Theorems & Definitions (37)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Lemma 3.1
  • ...and 27 more