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McKinsey-Tarski algebras and Raney extensions

Guram Bezhanishvili, Ranjitha Raviprakash, Anna Laura Suarez, Joanne Walters-Wayland

Abstract

We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the $T_0$-hull of a Raney extension generalizes that of the $T_D$-hull of a frame.

McKinsey-Tarski algebras and Raney extensions

Abstract

We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the -hull of a Raney extension generalizes that of the -hull of a frame.

Paper Structure

This paper contains 5 sections, 26 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Frames and MT-algebras
  3. MT-algebras and Raney extensions
  4. Raney morphisms between MT-algebras
  5. Equivalence of $\mathbf{MT_R}\xspace$ and $\mathbf{RE}\xspace$

Key Result

Theorem 2.2

$\mathbf{MT}\xspace$ is isomorphic to the category whose objects are pairs $(B,L)$ where $B$ is a complete boolean algebra and $L$ is a subframe of $B$ and whose morphisms are complete boolean morphisms $f: B \to B'$ such that the restriction $f : L \to L'$ is well defined. $\blacktriangleleft$$\bla

Theorems & Definitions (58)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 48 more