On systems of word equations with simple loop sets

Then the word d = v −k
1 x 0 is a prefix of both v 1 and v 2 . Surely
From (1) we have
|u k 1 | < min{|v 1|, |v 2 |}.
du i 1 x 1u i 2 x 2 · · ·u i m x m = v i−k
1 v i 2 y 2 · · ·v i n y n (i = k, k + 1, k + 2) . (4)
Let z 1 and z 2 be words such that
v 1 = du k 1 z 1 and v 2 = du k 1 z 2.
x 0 d u k
1
By (4),
v k 1
v k 1
v 1 z 1
v 2 z 2
u i 1x 1 u i 2 · · ·u i mx m = (u k 1z 1 d) i−k (u k 1z 2 d) i−1 u k 1z 2 y 2 v i 3y 3 · · ·v i ny n (5)
for i = k, k + 1, k + 2.
d u k 1 u 2 1
z 1 d u k
1
z 1 d u k
1
z 2
Consider the common prefix of u k+2
1 and (u k 1z 1 d) 2 u k 1.
If
|u k+2
1 | > |u 1 | + |u k 1 z 1d| − 1
then, by the Periodicity Lemma, the words u 1 and u k 1 z 1d have the same primitive
root t. Since v 1 and v 2 have primitive roots of equal length |t| and their
common prefix is longer than t, the words v 1 and v 2 commute.
Assume that
|v 1 | = |u k 1 z 1d| > |u k+1
1 | + 1.
If x 1 ≠ ǫ, then the words u 1 and x 1 are not marked.
d u k 1
x 1
d u k 1
u 1
v 1
Suppose therefore that x 1 = ǫ. Then (5) implies that u k+2
1 is a prefix of u k 1 z 1du 1 ,
and u k 1z 1 du 1 is comparable with u k+1
1 u 2 . Therefore also u k+2
1 and u k+1
1 u 2 are
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