FINITE SEMILATTICES WHOSE NON-INVERTIBLE ENDOMORPHISMS ARE PRODUCTS OF IDEMPOTENTS

For a finite semilattice S, is is proved that if every noninvertible e ndomorphism is a product of idempotents, then S is a chain; the conver se was proved, independently, by A. Ya. Aizenstat and J. M. Howie. For a finite pseudocomplemented semilattice S, with pseudocomplementation regarded as a unary operation, it is proved that all noninvertible en domorphisms are products of idempotents if and only if S is Boolean or a chain.