The structure of finite monoids satisfying the relation $\mathscr{R}=\mathscr{H}$

It is shown that any finite monoid $S$ on which Green's relations $\mathscr{R}$ and $\mathscr{H}$ coincide divides the monoid of all upper-triangular row-monomial matrices over a finite group. The proof is constructive; given the monoid $S$, the corresponding group and the order of matrices can be effectively found. The obtained result is used to identify the pseudovariety generated by all finite monoids satisfying $\mathscr{R}=\mathscr{H}$ with the semidirect product of the pseudovariety of all finite groups and the pseudovariety of all finite $\mathscr{R}$-trivial monoids.