It is well-known that some of the most basic properties of words, like the
commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed
as solutions of word equations. An important problem is to decide whether
or not a given equation on words has a solution. For instance,
the equation xMyN = zP
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, z are powers of w.
This result, which received a lot of attention, was first proved
by Lyndon and Schützenberger for free groups.
In this paper, we investigate equations on partial words.
Partial words are sequences over a finite alphabet that may contain
a number of “do not know” symbols. When we speak about equations
on partial words, we replace the notion of equality
(=) with compatibility (↑).
Among other equations, we solve xy ↑ yx,
xz ↑ zy, and special cases of xmyn ↑ zp
for integers m,n,p ≥ 2.