Idempotents and one-sided units : lattice invariants and a semigroup of functors on the category of monoids

For a monoid M, we denote by G(M) the group of units, E(M) the submonoid generated by the idempotents, and G_L(M) and G_R(M) the submonoids consisting of all left or right units. Writing M for the (monoidal) category of monoids, G, E, G_L and G_R are all (monoidal) functors M→M. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We show that this monoid has size 15, and describe its algebraic structure. We also show how to associate certain lattice invariants to a monoid, and classify the lattices that arise in this fashion. A number of examples are discussed throughout, some of which are essential for the proofs of the main theoretical results.