Word Equations I: Pairs and their Makanin-Razborov Diagrams
  This paper is the first in a sequence on the structure of sets of solutions to systems of equations over a free semigroup. To describe the structure, we present a Makanin-Razborov diagram that encodes the set of solutions to such system of equations. In the sequel we show how this diagram, and the tools that are used in constructing it, can be applied to analyze fragments of the first order theory of free semigroups. 
Z. Sela
math.GR math.LO math.RA