On the Burnside Semigroups xn = xn+m

In this paper we prove 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. De Luca and Varricchio solved the problem for n 5 in 90. A little later, 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 to eliminate all hypothesis 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 show that is a Church-Rosser rewriting system, thus solving the word problem, and show that the semigroup is nite J-above. We prove that the frame of the R-classes of the semigroup is a tree. We characterize and calculate the R-classes, H-classes and the D-classes of the semigroup, regular or not, and prove that its maximal subgroups are cyclic of order m whenever all productions of are stable. Recently Guba extended the cases in which the conjecture holds to n 3 and m 1. Using his work we obtain the stability of the productions of for n = 3 and m 1 too and, hence, all properties about the semigroup structure hold in this case.