Thue-Morse words and the \(\mathcal D\)-structure of the free Burnside semigroup (Q2755752)

Let \(\Sigma=\{a,b\}\), \(\Sigma^+\) be the free semigroup over \(\Sigma\), \(f\colon\Sigma^+\to\Sigma^+\) be the endomorphism such that \(f(a)=ab\) and \(f(b)=ba\). The Thue-Morse words are \(u_k=f^k(a)\) for \(k\geq 0\). So NEWLINE\[NEWLINEu_0=a,\quad u_1=ab,\quad u_2=abba,\quad u_3=abbabaab,\dots.NEWLINE\]NEWLINE Let \(\sim\) denote the verbal congruence of \(\Sigma^+\) corresponding to the semigroup variety defined by the identity \(x^2=x^3\) and \(B_2\) be the quotient semigroup of \(\Sigma^+\) by \(\sim\). Then \(B_2\) is the free two-generated semigroup in this variety.NEWLINENEWLINENEWLINEThe authors investigate certain properties of \(B_2\). The main results are the following: 1. The elements \(\widetilde u_k\) (\(k\geq 0\)) of \( B_2\) are pairwise different and non-regular. 2. The \(\mathcal D\)-classes of the elements \(\widetilde u_k\) (\(k\geq 0\)) of \(B_2\) are trivial and they form a chain of the kind \(\omega^*\) in the partially ordered set of \(\mathcal D\)-classes of \(B_2\).