Presentations of the Schützenberger Product of n Groups

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.

Key Words:

• Presentation
• Schützenberger product
• Semigroup

Mathematics Subject Classification:

• 20M05
• 20M35
• 20H25
• 68Q70

ACKNOWLEDGMENTS

This work was supported by the project POCTI/32440/MAT/2000 and “Financiamento Programático” of CAUL, by FEDER and by the INTAS project 1224.

We would like to thank the anonymous referee for its numerous suggestions.

Notes

¹Recall that the content of a word is the set of letters occurring in it.

Communicated by P. Higgins.