Characterization of the pseudovariety generated by finite monoids satisfying $\mathscr{R}=\mathscr{H}$

We consider the pseudovariety generated by all finite monoids on which Green's relations $\mathscr{R}$ and $\mathscr{H}$ coincide. It is shown that any finite monoid $S$ belonging to this pseudovariety divides the monoid of all upper-triangular row-monomial matrices over a finite group with zero adjoined. The proof is constructive; given a monoid $S$, the corresponding group and the order of matrices can be effectively found.