The Gumm level equals the alvin level in congruence distributive varieties

摘要

Congruence modular and congruence distributive varieties can be characterized by the existence of sequences of Gumm and Jonsson terms, respectively. Such sequences have variable lengths, in general. It is immediate from the above paragraph that there is a variety with Gumm terms but without Jonsson terms. We prove the unexpected result that, on the other hand, if some variety has both kinds of terms, then the minimal lengths of the sequences differ at most by 1. It follows that every congruence distributive variety with r+1 Day terms has r(2)-r+3 Jonsson terms.