On systems of word equations with simple loop sets

AbstractConsider the infinite system S of word equations {x0u1ix1u2ix2⋯umixm=y0v1iy1v2iy2⋯vniyn∣i∈N}. For each k∈N, let Tk 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 Tk such that primitive roots of φ(u1),φ(u2),…,φ(um) are of equal length, as well as primitive roots of φ(v1),φ(v2),…,φ(vn), then φ is a solution of the whole S.(2)If n=1 then, for any k≥2, a solution φ of Tk is also a solution of S