  We apply Stone duality and model theory to study the structure theory of free
pro-aperiodic monoids. Stone duality implies that elements of the free
pro-aperiodic monoid may be viewed as elementary equivalence classes of
pseudofinite words. Model theory provides us with saturated words in each such
class, i.e., words in which all possible factorizations are realized. We give
several applications of this new approach, including a solution to the word
problem for $\omega$-terms that avoids using McCammond's normal forms, as well
as new proofs and extensions of other structural results concerning free
pro-aperiodic monoids.
