On the

In this paper we show that the congruence classes of A associated to the Burnside semigroup with jAj generators deened by the equation x n = x n+m , for n 4 and m 1, are recognizable. This problem was originally formulated by Brzozowski in 1969 for m = 1 and n 2. Two years ago Aldo de Luca and Stefano Varricchio solved the problem for n 5. A little later, John McCammond extended the problem for m 1 and solved it independently in the cases n 6 and m 1. Our work, which is based on the techniques developed by de Luca and Varricchio, extends both these results. We eeectively construct a minimal generator of our congruence. We introduce an elementary concept, namely the stability of productions, which allows us to eliminate all hypotheses related to the values of n and m. A substantial part of our proof consists of showing that all productions in are stable for n 4 and m 1. We also give an algorithm that solves the word problem and shows that the semigroup is nite J-above. We prove that the frame of the R-classes of the semigroup is a tree. We characterize also the R-classes and the D-classes of the semigroup and prove that its maximal subgroups are cyclic of order m in the cases n 4 and m 1.