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Local compactness in MT-algebras

Guram Bezhanishvili, Ranjitha Raviprakash

Abstract

In our previous work, we introduced McKinsey-Tarski algebras (MT-algebras for short) as an alternative pointfree approach to topology. Here we study local compactness in MT-algebras. We establish the Hofmann-Mislove theorem for sober MT-algebras, using which we develop the MT-algebra versions of such well-known dualities in pointfree topology as Hofmann-Lawson, Isbell, and Stone dualities. This yields a new perspective on these classic results.

Local compactness in MT-algebras

Abstract

In our previous work, we introduced McKinsey-Tarski algebras (MT-algebras for short) as an alternative pointfree approach to topology. Here we study local compactness in MT-algebras. We establish the Hofmann-Mislove theorem for sober MT-algebras, using which we develop the MT-algebra versions of such well-known dualities in pointfree topology as Hofmann-Lawson, Isbell, and Stone dualities. This yields a new perspective on these classic results.
Paper Structure (10 sections, 69 theorems, 22 equations, 3 tables)

This paper contains 10 sections, 69 theorems, 22 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frames
  4. MT-algebras
  5. Lower separation axioms in MT-algebras
  6. Sober MT-algebras and Hofmann-Mislove theorem
  7. Locally compact MT-algebras and Hofmann-Lawson duality
  8. Stably locally compact and stably compact MT-algebras
  9. Locally compact Hausdorff MT-algebras
  10. Zero-dimensional MT-algebras

Key Result

Theorem 2.1

$$

Theorems & Definitions (145)

  • Theorem 2.1: see, e.g., PicadoPultr2012
  • Definition 2.2: McKinsey-Tarski algebras
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 135 more