The pseudovariety of semigroups of triangular matrices over a finite field

We show that semigroups representable by triangular matrices over a fixed finite field form a decidable pseudovariety and provide a finite pseudoidentity basis for it. Background and motivation The main results of this paper were motivated by one of the fundamental theorems of Imre Simon, namely, by his elegant algebraic characterization of the class of piecewise testable languages [21, 22]. This celebrated theorem was one of the main illuminating examples for the creation of the theory of pseudovarieties of finite semigroups and varieties of recognizable languages. By now there are a number of proofs [1, 11, 12, 23, 26] based on different approaches whose sources range from fairly concrete calculations in finite transformation semigroups to highly abstract constructions of model theory or profinite topology and so it has become a crossing where various profound ideas and techniques meet. Thus Simon’s Theorem has motivated a generation of researchers who have studied the relationship between finite semigroup theory and theoretical computer science. There are highly non-trivial purely algebraic consequences of Simon’s Theorem. Straubing [24] proved that a finite semigroup is J -trivial if and only if, for some n, it divides the semigroup Un of all n × n upper triangular Boolean matrices with all 1’s on the main diagonal. While it is easy to check that the semigroup Un is J -trivial, it is very difficult to prove that conversely every finite J -trivial semigroup divides Un for some n. The original proof in [24] ∗This work was completed in December 2003 when the third named author was visiting the University of Porto with the support of F.C.T. through Centro de Matemática and the project POCTI/32817/MAT/2000. He was also supported by the Science and Education Ministry of Russian Federation, grants E02-1.0-143 and 2227.2003.01. The first named author also acknowledges support of F.C.T. through Centro de Matemática and the project POCTI/32817/MAT/2000, which is funded, in part, by FEDER. The second named author was supported by the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israeli Science Foundation and by the Binational Science Foundation of the USA and Israel, grant 1999298/1. All three authors acknowledge the support from INTAS through the Network project 99-1224.