On the Pseudoperiodic Extension of u `

We investigate the solution set of the pseudoperiodic extension of the classical Lyndon and Schützenberger word equations. Consider u1 · · ·u` = v1 · · · vmw1 · · ·wn, where ui ∈ {u, θ(u)} for all 1 ≤ i ≤ `, vj ∈ {v, θ(v)} for all 1 ≤ j ≤ m, wk ∈ {w, θ(w)} for all 1 ≤ k ≤ n and u, v and w are variables, and θ is an antimorphic involution. A solution is called pseudoperiodic, if u, v, w ∈ {t, θ(t)}+ for a word t. Czeizler et al. (2011) established that for small values of `,m, and n non-periodic solutions exist, and that for large enough values all solutions are pseudoperiodic. However, they leave a gap between those bounds which we close for a number of cases. Namely, we show that for ` = 3 and either m,n ≥ 12 or m,n ≥ 5 and either m and n are not both even or not all ui’s are equal, all solutions are pseudoperiodic. 1998 ACM Subject Classification F.4.m Mathematical Logic and Formal Languages – Misc.