Presentations of the Schützenberger Product of n Groups

Abstract

ABSTRACT In this article, 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 B n (k). We show that B n (k) splits as a semidirect product of the monoid of unitriangular matrices U n (k) by the group of diagonal matrices. When the semiring is a field, B n (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 n (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 n × n unitriangular Boolean matrices.