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

TL;DR

This paper investigates the algebraic structure of functors related to idempotents and units in monoids, revealing a monoid of size 15 and classifying associated lattice invariants.

Contribution

It introduces a new monoid formed by functors on monoids related to idempotents and units, and classifies lattice invariants arising from these structures.

Findings

The monoid generated by these functors has size 15.

Classification of lattices associated with monoids.

Examples illustrating the theoretical results.

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.