Square-Free Words over Partially Commutative Alphabets

Abstract

There exist many constructions of infinite words over three-letter alphabet avoiding squares. However, the characterization of the lexicographically minimal square-free word is an open problem. Efficient construction of this word is not known. We show that the situation changes when some letters commute with each other. We give two characterizations (morphic and recursive) of the lexicographically minimal square-free word $\widetilde{\mathbf {v}}$ in the case of a partially commutative alphabet $\Theta$ of size three. We consider the only non-trivial relation of partial commutativity, for which $\widetilde{\mathbf {v}}$ exists: there are two commuting letters, while the third one is blocking (does not commute at all). We also show that the $n$-th letter of $\widetilde{\mathbf {v}}$ can be computed in time logarithmic with respect to $n$.