Monoids of upper triangular boolean matrices

10 pagesInternational audienceWe study the variety W generated by monoids of upper-triangular boolean matrices. First, we present W as a natural extension of the variety J of finite J-trivial monoids and we give a description of the family of recognizable languages whose syntactic monoids are in W. Then we show that W can be described in terms of the generalized Schützenberger product of finite monoids. We also show that W is generated by the power monoids of members of J. Finally we consider the membership problem for W and the connection with the dot-depth hierarchy in language theory. Although the majority of our results are purely "semigroup-theoretic" we use recognizable languages constantly in the proofs