On systems of word equations with simple loop sets

Consider the infinite system S of word equations

For each , let T_k be the subsystem of S given by i{k,k+1,k+2}. We prove two properties of the above system.

(1) Let k≥1. If φ is a solution of T_k such that primitive roots of are of equal length, as well as primitive roots of , then φ is a solution of the whole S.

(2) If n=1 then, for any k≥2, a solution φ of T_k is also a solution of S.

Keywords: Combinatorics on words; Equivalent subsystems; Pumping property