Andrey N. Plyushchenko and Arseny M. Shur
Abstract. In this paper we examine Brzozowski’s conjecture for the two-generated free Burnside semigroup satisfying x² = x³. The elements of this semigroup are classes of equivalent words, and the conjecture claims that all elements are regular languages. The case of the identity x² = x³ is the only one, for which Brzozowski’s conjecture is neither proved nor disproved. We prove the conjecture for all the elements containing an overlap-free or an “almost” overlap-free word. In addition, we show that all but finitely many of these elements are “big” languages in terms of growth rate.