Mitschke's Theorem is sharp

TL;DR

This paper demonstrates the optimal bounds for the number of Jönsson and Day terms in varieties with an m-ary near-unanimity term, establishing the sharpness of Mitschke's theorem and related congruence identities.

Contribution

It proves the bounds in Mitschke's theorem are optimal and characterizes the best possible bounds for various congruence identities in such varieties.

Findings

Mitschke's bounds are sharp for Jönsson and Day terms.

Characterization of exact bounds for congruence identities.

Optimal bounds for varieties with m-ary near-unanimity terms.

Abstract

A. Mitschke showed that a variety with an $m$-ary near-unanimity term has J\'onsson terms $t_0, \dots, t _{2m-4} $ witnessing congruence distributivity. We show that Mitschke's result is sharp. We also evaluate the best possible number of Day terms witnessing congruence modularity. More generally, we characterize exactly the best bounds for many congruence identities satisfied by varieties with an $m$-ary near-unanimity term.