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Day's Theorem is sharp for $n$ even

Paolo Lipparini

Abstract

We solve some problems about relative lengths of Maltsev conditions, in particular, we give an affirmative answer to a classical problem raised by A. Day more than fifty years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms $t_0, \dots, t_n$ witnessing congruence distributivity it is possible to construct terms $u_0, \dots, u _{2n-1} $ witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when $n$ is even. We also deal with other kinds of terms, such as alvin, Gumm, Pixley, directed, specular, mixed and defective. All the results hold when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.

Day's Theorem is sharp for $n$ even

Abstract

We solve some problems about relative lengths of Maltsev conditions, in particular, we give an affirmative answer to a classical problem raised by A. Day more than fifty years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms witnessing congruence distributivity it is possible to construct terms witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when is even. We also deal with other kinds of terms, such as alvin, Gumm, Pixley, directed, specular, mixed and defective. All the results hold when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.

Paper Structure

This paper contains 10 sections, 35 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. A review of congruence distributive and modular varieties
  3. The main constructions
  4. Day's Theorem is optimal for $n$ even
  5. Optimal bounds for varieties with directed terms
  6. Specular Maltsev conditions
  7. Gumm, directed Gumm and defective alvin terms
  8. Mixed Jónsson terms and congruence modularity
  9. Some more explicit descriptions and summing up everything
  10. Further remarks

Key Result

Theorem \oldthetheorem

(A. Day D) If $n>0$, then every $n$-distributive variety is $2n{-}1$-modular.

Theorems & Definitions (119)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
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  • Remark \oldthetheorem
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  • Remark \oldthetheorem
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  • ...and 109 more