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.
Categories: math.RA