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

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