The filter of interpretability types of Hobby-McKenzie varieties is prime

TL;DR

This paper investigates Hobby-McKenzie varieties within algebraic structures, providing new characterizations and proving that their interpretability types form a prime filter in the lattice of varieties.

Contribution

It introduces novel characterizations of Hobby-McKenzie varieties using compatible reflexive ternary structures and establishes the primeness of their interpretability filter.

Findings

New characterizations of Hobby-McKenzie varieties

Proof that the filter of their interpretability types is prime

Enhanced understanding of the structure of interpretability lattices

Abstract

We study the Hobby-McKenzie varieties that constitute a major class investigated thoroughly in the monograph The shape of congruence lattices by Kearnes and Kiss. We obtain new characterizations of the Hobby-McKenzie varieties via compatible reflexive ternary structures. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the interpretability types of Hobby-McKenzie varieties is prime.