Avoidance bases for formulas with reversal

Abstract

In the interest of studying formulas with reversal of high avoidability index, we find $n$-avoidance bases for formulas with reversal for $n\in\{1,2,3\}$. We demonstrate that there is a unique formula with reversal in each of these three bases of highest avoidability index $n+2$; these formulas are $xx$, $xyx\cdot y^R$, and $xyzx\cdot y^R\cdot z^R$, which belong to an infinite family of formulas with reversal that has been the subject of recent study by the authors.