Congruence lattices forcing nilpotency

TL;DR

This paper characterizes which congruence lattices determine nilpotency, supernilpotency, or solvability in algebraic structures, by analyzing possible commutator operations on these lattices.

Contribution

It introduces the concept of congruence lattices forcing nilpotency and describes conditions under which they enforce algebraic properties.

Findings

Identifies congruence lattices that force nilpotency, supernilpotency, or solvability.

Analyzes possible commutator operations on given congruence lattices.

Provides criteria for when a lattice enforces specific algebraic properties.

Abstract

Given a lattice $\mathbb{L}$ and a class $K$ of algebraic structures, we say that $\mathbb{L}$ \emph{forces nilpotency} in $K$ if every algebra $\mathbf{A} \in K$ whose congruence lattice $\mathrm{Con} (\mathbf{A})$ is isomorphic to $\mathbb{L}$ is nilpotent. We describe congruence lattices that force nilpotency, supernilpotency or solvability for some classes of algebras. For this purpose, we investigate which commutator operations can exist on a given congruence lattice.