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

Abstract

For a monoid $M$, we denote by $\mathbb{G}(M)$ the group of units, $\mathbb{E}(M)$ 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 $\mathcal{M}\to \mathcal{M}$. 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.

V2: referee suggestions incorporated, to appear in J Algebra. 23 pages, 12 figures, 4 tables