give the analogous definition for directed graphs and define stable directed graphs, which form a class between balanced and preordered directed graphs. Section 3 contains our main result, Theorem 3.2, which provides conditions for a reflexive directed graph to be the compression of a preordered directed graph. The proof of Theorem 3.2 takes up all of section 4. Theorem 3.2 also has a direct application in algebra (see [3]). 1 Compression Maps The directed graphs we consider have a finite number of vertices and no repeated arrows. Loops are allowed (a loop is an arrow from a vertex to itself). The vertex set and the arrow set of a directed graph D are denoted by V ( D ) and A ( D ), respectively. An arrow from vertex v to vertex w is denoted by vw . The notation D ∗ is reserved for the subgraph of D with vertex set V ( D $^{∗}$) = V ( D ) and arrow set A ( D $^{∗}$) = { xy ∈ A ( D ) : x,y ∈ V ( D ) and x = y } . We say a directed graph D is reflexive if vv ∈ A ( D ) for all v ∈ V ( D ) and transi- tive if xy,yz ∈ A ( D ) implies xz ∈ A ( D ) for all x,y,z ∈ V ( D ). If D is reflexive and transitive then we say D is preordered . A transitive triple in D is an ordered triple of vertices contained in Trans ( D ) = { ( a,b,c ) : a,b,c ∈ V ( D ) and ab,bc,ac ∈ A ( D ) } . Definition 1.1 Suppose D$_{1}$ and D$_{2}$ are reflexive directed graphs. A compression map is a surjective function θ : V ( D$_{2}$ ) → V ( D$_{1}$ ) which satisfies 1, 2, and 3 below. 1. θ ( x ) θ ( y ) ∈ A ( D$_{1}$ ) for all x,y ∈ D$_{2}$ such that xy ∈ A ( D$_{2}$ ) . 2