Normal submonoids and congruences on a monoid

TL;DR

This paper introduces the concept of normal submonoids in monoids, explores their lattice structure, and connects them to congruences, providing new methods for computing congruences especially in transformation monoids.

Contribution

It generalizes normal subgroups to monoids, describes their lattice structure, and links them to congruences, offering a new approach for computing congruences on monoids.

Findings

The set of normal submonoids forms a complete lattice.

Explicit descriptions of joins in the lattice are provided.

The lattice of normal submonoids embeds into the lattice of congruences.

Abstract

A notion of {\em normal submonoid} of a monoid $M$ is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf{NorSub}(M)$ of normal submonoids of $M$ is a complete lattice. Joins are explicitly described, and the lattice is computed for the finite full transformation monoids $T_n$, $n\geq 1$. It is also shown that $\mathsf{NorSub}(M)$ is modular for a specific family of commutative monoids, including all Krull monoids, and that, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice $\mathsf{Cong}(M)$ of congruences on $M$. This leads to a new strategy for computing $\mathsf{Cong}(M)$ consisting of computing $\mathsf{NorSub}(M)$, and the lattices of the so called unital congruences on the quotients of $M$ modulo its normal submonoids. This provides a new perspective on Malcev computation of the congruences on $T_n$.