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Presentations of the Schützenberger product of n groups

Gomes, Gracinda M. S.; Pin, J.-E.; Sezinando, Helena

Communications in Algebra, 34 (4) (2006), 1213-1235

In this paper, we first consider nxn upper-triangular matrices with entries in a given semiring k.
Matrices of this form with invertible diagonal entries form a monoid B(k). We show that this monoid splits as a semidirect product of the monoid of unitriangular matrices U(k) by the group of diagonal matrices. When the semiring is a field, B(k) is actually a group
and we recover a well-known result from the theory of groups and Lie algebras. Pursuing the analogy with the group case, we show that U(k)is the ordered set product of n(n-1)/2 commutative monoids (the root subgroups in the group case).
Finally, we give two different presentations of the Schützenberger product of n groups $G_1$,...,$G_n$, given a monoid presentation <A_i|R_i> of each group $G_i$. We also obtain as a special case
presentations for the monoid of all nxn unitriangular Boolean matrices.