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Finitely Based Congruence Varieties

Ralph Freese, Paolo Lipparini

Abstract

We show that for a large class of varieties of algebras, the equational theory of the congruence lattices of the members is not finitely based.

Finitely Based Congruence Varieties

Abstract

We show that for a large class of varieties of algebras, the equational theory of the congruence lattices of the members is not finitely based.
Paper Structure (7 sections, 22 theorems, 23 equations, 2 figures)

This paper contains 7 sections, 22 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Haiman's lattices and higher order Arguesian identities
  4. Projectivity of $\mathbf{M}_3$ in congruence varieties
  5. Proof of Theorem \ref{['thm:main']}($\operatorname{1}'$)
  6. Lattices in $\textit{\bfseries\slshape{S}}\,\mathbf{Con}(\mathscr V)$
  7. Proofs of Theorem \ref{['thm:main']}($\operatorname{2}'$) and Theorem \ref{['thm:fbcv']}

Key Result

Theorem 1.1

There is no nontrivial finitely based modular congruence variety other than the variety of distributive lattices.

Figures (2)

  • Figure 1:
  • Figure 2: Two members of $\mathscr K_\infty$

Theorems & Definitions (37)

  • Theorem 1.1: Freese1994
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 27 more