Me. Adams et al., FINITE SEMILATTICES WHOSE NON-INVERTIBLE ENDOMORPHISMS ARE PRODUCTS OF IDEMPOTENTS, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 124, 1994, pp. 1193-1198
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.