Commutative nilpotent transformation semigroups

Cameron et al. determined the maximum size of a null subsemigroup of the full transformation semigroup \(\mathcal {T}(X)\) on a finite set X and provided a description of the null semigroups that achieve that size. In this paper we extend the results on null semigroups (which are commutative) to commutative nilpotent semigroups. Using a mixture of algebraic and combinatorial techniques, we show that, when X is finite, the maximum order of a commutative nilpotent subsemigroup of \(\mathcal {T}(X)\) is equal to the maximum order of a null subsemigroup of \(\mathcal {T}(X)\) and we prove that the largest commutative nilpotent subsemigroups of \(\mathcal {T}(X)\) are the null semigroups previously characterized by Cameron et al.