3. Suppose ( s, b, c ) ∈ Trans ( V ( D )) for some b, c ∈ V ( D ) \ { s } with b = c . (a) If cd ∈ A ( D ) for some vertex d then bd ∈ A ( D ) if and only if sd ∈ A ( D ) . (b) If as ∈ A ( D ) for some vertex a then ab ∈ A ( D ) if and only if ac ∈ A ( D ) . Proof. The proofs of parts 1, 2, and 3 are similar. Note that in part 2 we have a = c since s is a soloist. We prove part 1. Assume ( a, b, s ) ∈ Trans ( D ) for some a, b ∈ V ( D ) \ { s } with a = b . (a) Suppose sc ∈ A ( D ) for some c ∈ V ( D ) \ { s } . If ac ∈ A ( D ) then applying the balance property to a, b, s, c gives bc ∈ A ( D ). On the other hand if bc ∈ A ( D ) then a = c and b = c since s is a soloist. Applying the stable property to a, b, s, c gives ac ∈ A ( D ). The proof of Theorem 3.2 is a constructive algorithm described in section 4. In each iteration of the algorithm we construct a preordered directed graph with one more vertex and define a compression. The algorithm stops when we arrive at a preordered directed graph and the desired compression is obtained by composition. We finish this section with an example which covers the steps and constructions given in the proof of Theorem 3.2. The directed graphs in Figure 4 are stable. We may identify (i) as a compression of (ii) by mapping 2 and t$_{1}$ to 2. We may also identify (ii) as a compression of (iii) by mapping 4 and t$_{2}$ to 4. (b) Suppose xa ∈ A ( D ) for some x ∈ V ( D ) \ { a, b } . If xs ∈ A ( D ) then applying the balance property to x, a, b, s gives xb ∈ A ( D ). On the other hand if xb ∈ A ( D ) then x = s since s is a soloist. Applying the stable property to x, a, b, s gives xs ∈ A ( D ).