Finite structurally uniform groups and commutative nilsemigroups

Abstract

Let $S$ be a finite semigroup. By $\mathrm{S}\mathrm{u}\mathrm{b}(S)$ we denote the lattice of all its subsemigroups. If $A \in \mathrm{S}\mathrm{u}\mathrm{b}(S)$, then by $h(A)$ we denote the height of the subsemigroup $A$ in the lattice $\mathrm{S}\mathrm{u}\mathrm{b}(S)$. A semigroup $S$ is called structurally uniform if, for any $A, B \in \mathrm{S}\mathrm{u}\mathrm{b}(S)$ the condition $h(A) = h(B) implies that A \sim = B$. We present a classification of finite structurally uniform groups and commutative nilsemigroups.