ABSTRACT
As we saw in Chapter 2, some of the most basic properties of words, like the conjugacy and the commutativity, can be expressed as solutions of word equations. Recall that two words x and y are conjugate if there exist words u and v such that x = uv and y = vu. The latter is equivalent to the existence of a word z satisfying xz = zy in which case there exist words u, v such that x = uv, y = vu, and z = (uv)ku for some nonnegative integer k. And two words x and y commute, namely xy = yx, if and only if x and y are powers of the same word, that is, there exists a word z such that x = zk and y = zl
for some integers k and l. Another equation of interest is xm = yn. It turns out that if x and y are
words, then xm = yn for some positive integers m,n if and only if there exists a word z such that x = zk and y = zl for some integers k and l. Yet, another interesting equation is xmyn = zp which has only periodic solutions in a free monoid, that is, if xmyn = zp holds with integers m,n, p ≥ 2, then there exists a word w such that x, y and z are powers of w. In this chapter, we pursue our investigation of equations on partial words.
In Section 10.1, we give a result that gives the structure of partial words satisfying the equation xm ↑ yn, which provides the conditions for when x and y are contained in powers of a common word. In Section 10.2, we solve the equation x2 ↑ ymz. This result is a first step for solving the equation xmyn ↑ zp in Section 10.3.
