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On the preservation of unification type of Heyting algebras and interior algebras

Ivo Düntsch, Wojciech Dzik

Abstract

The purpose of this note is to shed some light on the preservation of unification types of locally finite varieties of interior algebras and varieties of Heyting algebras under the functors presented by W. Blok in his dissertation.

On the preservation of unification type of Heyting algebras and interior algebras

Abstract

The purpose of this note is to shed some light on the preservation of unification types of locally finite varieties of interior algebras and varieties of Heyting algebras under the functors presented by W. Blok in his dissertation.

Paper Structure

This paper contains 9 sections, 30 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and first definitions
  3. Algebraic unification
  4. Interior algebras and Heyting algebras
  5. Preservation of unification type
  6. From $\mathbb{HA}$ to $\mathbb{IA}$
  7. From $\mathbb{IA}$ to $\mathbb{HA}$
  8. Summary and outlook
  9. References

Key Result

Lemma 3.1

Suppose that $M$ is a minimal algebra in the variety $\mathbf{V}$ and projective in $\mathbf{V}$. Then, $A \in \mathbf{V}\xspace$ is unifiable if and only if $M$ is a homomorphic image of $A$.

Figures (3)

  • Figure 1: Quasiordering unifiers
  • Figure 2: The functor $\mathcal{O}\xspace$
  • Figure 3: The functor $\mathcal{B}\xspace$

Theorems & Definitions (49)

  • Lemma 3.1
  • proof
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Corollary 4.3
  • proof
  • proof
  • Lemma 4.4
  • proof
  • ...and 39 more