Let c ∈ Supp$_{G}$ S and g ∈ S$_{c}$ \ { 0 } be arbitrarily chosen. Since h ( g ) ∈ T \ { 0 } there exist x$_{2}$, y$_{2}$ ∈ X$_{2}$ such that x$_{2}$ρ$_{2}$y$_{2}$ and ( h ( g )) ( x$_{2}$, y$_{2}$ ) = 0. If we set a$_{2}$ = θ ( x$_{2}$ ) and b$_{2}$ = θ ( y$_{2}$ ) then a$_{2}$ρ$_{1}$b$_{2}$ since θ preserves order. Moreover ( h ( g )) ( x$_{2}$, y$_{2}$ ) = g ( a$_{2}$, b$_{2}$ ) and Φ$_{1}$ ( a$_{2}$, b$_{2}$ ) = c since the grading on S is induced by Φ$_{1}$. We have Φ$_{2}$ ( x$_{2}$, y$_{2}$ ) = Φ$_{1}$ ( θ ( x$_{2}$ ) , θ ( y$_{2}$ )) so Φ$_{2}$ ( x$_{2}$, y$_{2}$ ) = c hence h ( g ) ∈ T$_{c}$ as desired. This proves h preserves the grading. Theorem 4.3 Suppose ρ$_{1}$ and ρ$_{2}$ are reflexive relations on X$_{1}$ and X$_{2}$ , respectively, θ is a compression map of X$_{2}$ onto X$_{1}$ , and G is a cancellative monoid. There is a G -grading set of ρ$_{1}$ if and only if there is a G -grading set of ρ$_{2}$ . Proof. We assume there is a G -grading set of ρ$_{2}$ and prove there is a G -grading set of ρ$_{1}$ . The reverse implication can be proved using a similar argument. Suppose σ$_{2}$ is a G -grading set of ρ$_{2}$ over G . Then σ$_{2}$ ⊆ ρ ∗ 2 since G is cancellative. We set σ$_{1}$ = { θ $^{∗}$( a, b ) : a, b ∈ X$_{1}$ and ( a, b ) ∈ σ$_{2}$ } . First we prove σ$_{1}$ is a G -essential set for ρ$_{1}$ . Suppose Φ$_{1}$ , Φ ′ 1 : ρ$_{1}$ → G are homomorphisms such that Φ$_{1}$ = Φ ′ $_{1}$. Then there exist a$_{1}$, a$_{2}$ ∈ X$_{1}$ such that a$_{1}$ρ ∗ $_{1}$a$_{2}$ and Φ$_{1}$ ( a$_{1}$, a$_{2}$ ) = Φ ′ $_{1}$( a$_{1}$, a$_{2}$ ). We have ( a$_{1}$, a$_{2}$ ) = θ $^{∗}$( x$_{1}$, x$_{2}$ ) for some x$_{1}$, x$_{2}$ ∈ X$_{2}$ since θ ∗ is bijective. Let Φ$_{2}$ : ρ$_{2}$ → G be the homomorphism defined as in part 1 of Lemma 4.2. Let Φ ′ 2 : ρ$_{2}$ → G be defined similarly, using Φ ′ 1 in place of Φ$_{1}$. We have Φ$_{2}$ ( x$_{1}$, x$_{2}$ ) = Φ$_{1}$ ( a$_{1}$, a$_{2}$ ) and Φ ′ $_{2}$( x$_{1}$, x$_{2}$ ) = Φ ′ $_{1}$( a$_{1}$, a$_{2}$ ) so Φ$_{2}$ ( x$_{1}$, x$_{2}$ ) = Φ ′ $_{2}$( x$_{1}$, x$_{2}$ ). Thus there exist y, z ∈ X$_{1}$ such that ( y, z ) ∈ σ$_{2}$ and Φ$_{2}$ ( y, z ) = Φ ′ $_{2}$( y, z ) since σ$_{1}$ is a G -grading set for ϱ$_{1}$ . We have θ $^{∗}$( y, z ) ∈ σ$_{1}$ , Φ$_{1}$ ( θ $^{∗}$( y, z )) = Φ$_{2}$ ( y, z ), and Φ ′ $_{1}$( θ $^{∗}$( y, z )) = Φ ′ $_{2}$( y, z ) by construction. Therefore Φ$_{1}$ ( θ $^{∗}$( y, z )) = Φ ′ $_{1}$( θ $^{∗}$( y, z )) and this shows σ$_{1}$ is a G -essential set for ρ$_{1}$ . 13