2. For all ( a$_{1}$, a$_{2}$, a$_{3}$ ) ∈ Trans ( D$_{1}$ ) there exists ( x$_{1}$, x$_{2}$, x$_{3}$ ) ∈ Trans ( D$_{2}$ ) such that θ ( x$_{i}$ ) = a$_{i}$ for i = 1 , 2 , 3 . 3. There is a bijection θ ∗ : A ( D ∗ $_{2}$) → A ( D ∗ $_{1}$) given by θ $^{∗}$( xy ) = θ ( x ) θ ( y ) for all x, y ∈ V ( D$_{2}$ ) with xy ∈ A ( D ∗ $_{2}$) . A figure showing a reflexive directed graph D will only display D and will not show the loops. Thus we assume the directed graphs in Figure 1 are both reflexive. In Example 1.2 we show directed graph (b) is a compression of directed graph (a). Figure 1: (a) is transitive and (b) is not transitive. Example 1.2 Let D$_{1}$ and D$_{2}$ be the reflexive directed graphs with V ( D$_{1}$ ) = { x, y, z } , A ( D ∗ $_{1}$) = { xy, yz } , V ( D$_{2}$ ) = { x, y, z, t } , and A ( D ∗ $_{2}$) = { xy, tz } where x, y, z, t are distinct. In Figure 1 we can match up D$_{1}$ with (a) and D$_{2}$ with (b). A compression map θ : V ( D$_{2}$ ) → V ( D$_{1}$ ) is given by θ ( x ) = x , θ ( y ) = y , θ ( z ) = z , and θ ( t ) = y . The effect on the directed graphs is to map the two middle vertices of (a) to the middle vertex of (b). Lemma 1.3 Let D$_{1}$ and D$_{2}$ be reflexive directed graphs and let θ : V ( D$_{2}$ ) → V ( D$_{1}$ ) be a compression. Suppose xy, yz ∈ A ( D$_{2}$ ) and ( θ ( x ) , θ ( y ) , θ ( z )) ∈ Trans ( D$_{1}$ ) for some x, y, z ∈ V ( D$_{2}$ ) . Then ( x, y, z ) ∈ Trans ( D$_{2}$ ) . Proof. Choose arbitrary x, y, z ∈ V ( D$_{2}$ ) such that xy, yz ∈ A ( D$_{2}$ ). If x, y, z are not distinct then xz ∈ A ( D$_{2}$ ) follows immediately. If x, y, z are distinct then set 3