A squarefree term not occurring in the Leech sequence

Abstract

Let \[ \begin{array}{c}\overline{A} = ABCBA\ CBC\ ABCBA,\\ \overline{B} = BCACB\ ACA\ BCACB,\\ \overline{C} = CABAC\ BAB\ CABAC. \end{array} \] The Leech sequence $L$ is the squarefree sequence obtained as the limit of the palindromes \[ A, \overline{A}, \overline{\overline{A}}, \ldots . \] In order to specify a certain class of pseudorecursive varieties of semigroups, it is helpful to have a squarefree term in 3 variables such that no substitution instance occurs as a subterm of $L$. We show that $\kappa_1 = aba\ cbc\ aba\ c$ is such a term. Except for one situation, the doubly-linked term $\kappa_2 = aba\ cbc\ aba$ will serve, and we focus on it.