Absorption and directed J\'{o}nsson terms

TL;DR

This paper proves that congruence distributive and modular varieties have directed Jönsson and Gumm terms, respectively, providing new insights into absorption properties and extending finite algebra results to general algebras.

Contribution

It establishes the existence of directed Jönsson and Gumm terms in all congruence distributive and modular varieties, respectively, generalizing previous finite algebra results.

Findings

Directed Jönsson terms exist in all congruence distributive varieties.

Directed Gumm terms exist in all congruence modular varieties.

New proof of Lipparini's result on Pixley terms in certain varieties.

Abstract

We prove that every congruence distributive variety has directed J\'{o}nsson terms, and every congruence modular variety has directed Gumm terms. The directed terms we construct witness every case of absorption witnessed by the original J\'{o}nsson or Gumm terms. This result is equivalent to a pair of claims about absorption for admissible preorders in CD and CM varieties, respectively. For finite algebras, these absorption theorems have already seen significant applications, but until now, it was not clear if the theorems hold for general algebras as well. Our method also yields a novel proof of a result by P. Lipparini about the existence a chain of terms (which we call Pixley terms) in varieties that are at the same time congruence distributive and $k$-permutable for some $k$.