Geodesic
The structure of finite monoids satisfying the relation $\mathscr{R}=\mathscr{H}$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 181-191

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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.
Keywords: finite monoids, Green’s relations, monoid representation, monoid pseudovariety, upper-triangular matrices. 
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     title = {The structure of finite monoids satisfying the relation $\mathscr{R}=\mathscr{H}$},
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T. V. Pervukhina. The structure of finite monoids satisfying the relation $\mathscr{R}=\mathscr{H}$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 181-191. http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a18/