Minimal clones with weakly abelian representations

TL;DR

This paper establishes a fundamental equivalence between the existence of nontrivial weakly abelian and abelian representations in minimal clones, and shows that all such representations are weakly abelian when they exist.

Contribution

It proves that minimal clones have nontrivial weakly abelian representations if and only if they have nontrivial abelian representations, and all such representations are weakly abelian.

Findings

Nontrivial weakly abelian representations exist iff abelian representations exist.

All representations in this case are weakly abelian.

Provides a characterization of minimal clones with weakly abelian representations.

Abstract

We show that a minimal clone has a nontrivial weakly abelian representation iff it has a nontrivial abelian representation, and that in this case all representations are weakly abelian.