On systems of word equations with simple loop sets

Consider the infinite system SS of word equations {x0u1ix1u2ix2⋯umixm=y0v1iy1v2iy2⋯vniyn∣i∈N}. For each k∈Nk∈N, let TkTk be the subsystem of SS given by i∈{k,k+1,k+2}i∈{k,k+1,k+2}. We prove two properties of the above system. (1)Let k≥1k≥1. If φφ is a solution of TkTk 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 φφ is a solution of the whole SS.(2)If n=1n=1 then, for any k≥2k≥2, a solution φφ of TkTk is also a solution of SS.