Presentations of the Schützenberger product of n groups

In this paper, we first consider n × n upper-triangular matrices with entries in a given semiring k. Matrices of this form with invertible diagonal entries form a monoid Bn(k). We show that Bn(k) splits as a semidirect product of the monoid of unitriangular matrices Un(k) by the group of diagonal matrices. When the semiring is a field, Bn(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 Un(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 G1, . . . , Gn, given a monoid presentation 〈Ai | Ri〉 of each group Gi. We also obtain as a special case presentations for the monoid of all n× n unitriangular Boolean matrices.