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Varieties of modal algebras without the congruence extension property

Zalán Gyenis, Zalán Molnár

Abstract

In a recent paper, Krawczyk proved that there are continuum many axiomatic extensions of global consequence associated with the modal system $E$ that do not admit the local deduction detachment theorem. In algebraic parlance, he showed that there are continuum many varieties of modal algebras lacking the congruence extension property. In this paper, we extend Krawczyk's results and construct a continuum of varieties of modal algebras that do not have the congruence extension property, but that do admit other, logically relevant properties, such as monotonicity, extensiveness, idempotency, normality, etc. This gives a continuum of axiomatic extensions of the corresponding modal systems not having the local deduction detachment theorem.

Varieties of modal algebras without the congruence extension property

Abstract

In a recent paper, Krawczyk proved that there are continuum many axiomatic extensions of global consequence associated with the modal system that do not admit the local deduction detachment theorem. In algebraic parlance, he showed that there are continuum many varieties of modal algebras lacking the congruence extension property. In this paper, we extend Krawczyk's results and construct a continuum of varieties of modal algebras that do not have the congruence extension property, but that do admit other, logically relevant properties, such as monotonicity, extensiveness, idempotency, normality, etc. This gives a continuum of axiomatic extensions of the corresponding modal systems not having the local deduction detachment theorem.
Paper Structure (7 sections, 26 theorems, 56 equations, 2 figures)

This paper contains 7 sections, 26 theorems, 56 equations, 2 figures.

Table of Contents

  1. Overview
  2. Constructions and proofs
  3. Extensive and contractive varieties without the CEP
  4. The subadditive case
  5. Monotone, extensive and idempotent
  6. The semi-complemented case
  7. The symmetric and extensive case

Key Result

Theorem Theorem \ref{THM:EXT}

There are continuum many extensive subvarieties of Boolean frames lacking the congruence extension property.

Figures (2)

  • Figure 1: Non-trivial congruences of $\mathfrak{B}$.
  • Figure 2: The frame $\mathcal{W}_9$.

Theorems & Definitions (27)

  • Theorem Theorem \ref{THM:EXT}
  • Corollary 1.1
  • Corollary 1.2
  • Theorem Theorem \ref{thm:main2}
  • Corollary 1.3
  • Theorem Theorem \ref{THM:COMP}
  • Corollary 1.4
  • Theorem Theorem \ref{thm:main3}
  • Corollary 1.5
  • Theorem 2.2
  • ...and 17 more