On the Burnside semigroups xn= xn+m

In this paper we show that the congruence classes of A* associated to the Burnside semigroup with ¦A¦ generators defined by the equation xn=xn+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 effectively 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 finite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{J}$$ \end{document}-above. We prove that the frame of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{R}$$ \end{document}-classes of the semigroup is a tree. We characterize also the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{R}$$ \end{document}-classes and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{D}$$ \end{document}-classes of the semigroup and prove that its maximal subgroups are cyclic of order m in the cases n≥4 and m≥1.