An extension of the Lyndon–Schützenberger result to pseudoperiodic words

One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson–Crick complement, denoted here as θ(u). Thus, any expression consisting of repetitions of u and θ(u) can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form ul=vnwm, to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if l⩾5,n,m⩾3, then all three words involved can be expressed in terms of a common word t and its complement θ(t). Moreover, if l⩾5, then n=m=3 is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement θ(u), which is also obtained in this paper.