Presentations of the Schützenberger product of n groups

Gracinda Gomes (1) , Jean-Eric Pin (2) , Helena Sezinando (1)

1 CAUL - Centro de Algebra da Universidade de Lisboa
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

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 of each group Gi. We also obtain as a special case presentations for the monoid of all n × n unitriangular Boolean matrices.

Mots clés: Schützenberger product; upper-triangular matrices; presentation of a monoid

Domaines: Théorie des groupes [math.GR]; Mathématique discrète [cs.DM]

Gracinda Gomes, Jean-Eric Pin, Helena Sezinando. Presentations of the Schützenberger product of n groups. Communications in Algebra, 2006, 340, pp.1213--1235. ⟨hal-00112839⟩