Higher Kiss Terms

TL;DR

This paper establishes the equivalence of two higher commutator concepts in modular varieties, introduces a procedure for generating higher dimensional congruences, and extends key theorems to this broader class.

Contribution

It proves the equality of the modular term condition higher commutator and the hypercommutator, and develops a method to produce higher dimensional Kiss terms in modular varieties.

Findings

HC8 holds for modular varieties

Every modular variety has an infinite sequence of higher dimensional Kiss terms

Extended Opršal's theorem from permutable to modular varieties

Abstract

We show that the modular term condition higher commutator is equal to the modular hypercommutator. As a consequence, we arrive at a new proof that HC8 holds for modular varieties. Next, we develop a procedure for a modular variety for producing the higher dimensional congruences that characterize the hypercommutator. This procedure allows us to demonstrate that every modular variety has an infinite sequence of what we call higher dimensional Kiss terms. We use these results to extend the scope of a theorem of Opr\v{s}al from permutable varieties to modular varieties.