define ˜ Υ$_{i}$ : ˜ ρ → G by ˜ Υ$_{i}$ ([ a ] , [ b ]) = Υ$_{i}$ ( a, b ) for all a, b ∈ P such that ( a, b ) ∈ ρ . A routine check proves ˜ Υ$_{1}$ , ˜ Υ$_{2}$ are homomorphisms such that ˜ Υ$_{1}$ |$_{˜}$ β = ˜ Υ$_{2}$ |$_{˜}$ $_{β}$. Thus ˜ Υ$_{1}$ = ˜ Υ$_{2}$ since ˜ β is a G -grading set for ˜ X . This proves Υ$_{1}$ ( a, b ) = Υ$_{2}$ ( a, b ) for all a, b ∈ P such that ( a, b ) ∈ ρ . Suppose x$_{1}$, x$_{2}$ ∈ X such that x$_{1}$ρx$_{2}$ are given. There are uniquely determined p$_{1}$, p$_{2}$ ∈ P such that [ x$_{1}$ ] = [ p$_{1}$ ] and [ x$_{2}$ ] = [ p$_{2}$ ]. Transitivity and the homomor- phism property gives Υ$_{1}$ ( x$_{1}$, x$_{2}$ ) = Υ$_{1}$ ( p$_{1}$, x$_{1}$ ) − $^{1}$Υ$_{1}$ ( p$_{1}$, p$_{2}$ ) Υ$_{1}$ ( p$_{2}$, x$_{2}$ ). By the result of the previous paragraph Υ$_{1}$ ( p$_{1}$, p$_{2}$ ) = Υ$_{2}$ ( p$_{1}$, p$_{2}$ ). Moreover Υ$_{1}$ ( p$_{2}$, x$_{2}$ ) = Υ$_{2}$ ( p$_{2}$, x$_{2}$ ) and Υ$_{1}$ ( p$_{1}$, x$_{1}$ ) = Υ$_{2}$ ( p$_{1}$, x$_{1}$ ) since Υ$_{1}$ |$_{σ}$ = Υ$_{2}$ |$_{σ}$ and ( p$_{1}$, x$_{1}$ ) , ( p$_{2}$, x$_{2}$ ) ∈ γ . Sub- stitution gives Υ$_{1}$ ( x$_{1}$, x$_{2}$ ) = Υ$_{2}$ ( p$_{1}$, x$_{1}$ ) − $^{1}$Υ$_{2}$ ( p$_{1}$, p$_{2}$ ) Υ$_{2}$ ( p$_{2}$, x$_{2}$ ) and this reduces to Υ$_{1}$ ( x$_{1}$, x$_{2}$ ) = Υ$_{2}$ ( x$_{1}$, x$_{2}$ ). Since x$_{1}$, x$_{2}$ ∈ X such that ( x$_{1}$, x$_{2}$ ) ∈ ρ were arbitrarily chosen we can conclude Υ$_{1}$ = Υ$_{2}$ and σ is a G -essential set for ρ . Therefore σ is a G -grading set for ρ . Corollary 5.2 Assume ( X, ρ ) is the compression of a preorder ( Y, ≤ ) . If Y has a cross-cut of length one or two then there is a subset σ of ρ such that σ is a G -grading set for ρ for any group G . Proof. This follows from Theorems 4.3 and 5.1. The directed graphs in Figure 2 represent reflexive relations. The vertices have been replaced with elements of the sets they represent. Example 5.3 Suppose ρ$_{1}$ is the stable relation on X$_{1}$ = { 1 , 2 , 3 , 4 } determined by (a) in Figure 2. The only clasp in X$_{1}$ is 2, which is an unlocked clasp, so X$_{1}$ is the compression of a preordered set by part 1 of Theorem 4.5. If ρ$_{2}$ is the reflexive 17