We prove that the congruence lattice of a nilsemigroup is modular if and only if the width of the semigroup, as a poset, is at most two, and distributive if and only if its width is one. In the latter case, such semigroups therefore coincide with the nil Δ">Δ Δ -semigroups. It is further shown that if a finitely generated nilsemigroup has modular congruence lattice, then the semigroup is finite.
Popovich, Alexander L. and Jones, Peter, "On Congruence Lattices of Nilsemigroups" (2017). Mathematics, Statistics and Computer Science Faculty Research and Publications. 525.