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
*x*^{ m}
y^{ n} = *z*^{ p} has only periodic solutions in a free monoid, that is, if
*x*^{ m}
y^{ n} = *z*^{ p} 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 *x*^{ m} y^{ n} ↑ *z*^{
p} for integers
*m, n, p* ≥ 2.