Normal submonoids and congruences on a monoid

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 $\mathsf{N}\mathsf{o}\mathsf{r}\mathsf{S}\mathsf{u}\mathsf{b}(M)$ of normal submonoids of $M$ is a complete lattice. Joins are explicitly described, and the lattice is computed for the finite full transformation monoids $T_n$, $n\ge 1$. It is also shown that $\mathsf{N}\mathsf{o}\mathsf{r}\mathsf{S}\mathsf{u}\mathsf{b}(M)$ 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 $\mathsf{C}\mathsf{o}\mathsf{n}\mathsf{g}(M)$ of congruences on $M$. This leads to a new strategy for computing $\mathsf{C}\mathsf{o}\mathsf{n}\mathsf{g}(M)$ consisting of computing $\mathsf{N}\mathsf{o}\mathsf{r}\mathsf{S}\mathsf{u}\mathsf{b}(M)$, 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 $T_n$.