On Brzozowski’s Conjecture for the Free Burnside Semigroup Satisfying x2 = x3
In this paper we examine Brzozowski’s conjecture for the two-generated free Burnside semigroup satisfying x 2 = x 3. 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 x 2 = x 3 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.
KeywordsWord Problem Regular Language Congruence Class Free Semigroup Primary Series
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