The class of nilpotent semigroups was introduced, via a semigroup identity, independently in  and , cf. , Every nilpotent cancellative semigroup S was shown to have a group of classical fractions G which is nilpotent and of the same nilpotency class as S. Groups, and linear groups in particular, satisfying certain related semigroup identities, introduced in , have been recently studied in ,  and . In particular, finitely generated residually finite groups satisfying a semigroup identity must be almost nilpotent, . On the other hand, it was recently shown in  that a finitely generated linear semigroup S ⊆ Mn(K), over a field K, with no free non-commutative subsemigroups satisfies an identity and for every maximal subgroup H of Mn(K) the subgroup gp(S ∩ H) of H generated by S ∩ H is almost nilpotent. A natural question that arises here is to decide which of these semigroups are nilpotent. Because of the powerful classical theory of nilpotent linear groups, cf. , one can also ask whether such semigroups can be approached via group theoretical methods. We note that the very special case of nilpotent connected algebraic monoids has been recently considered in . In  the structure of semigroup algebras of nilpotent semigroups was studied, in particular via prime Goldie homomorphic images, leading naturally to nilpotent subsemigroups of the matrix monoids Mn(D) over division rings D.