4 Compression Maps and Stable Relations Suppose Φ$_{1}$ , Φ$_{2}$ are homomorphism and Φ$_{1}$ |$_{σ}$ = Φ$_{2}$ |$_{σ}$ . Give x, y ∈ X such that xρy there is a unique chain { x$_{1}$, . . . , x$_{m}$ } in σ such that x = x$_{1}$ ≤ x$_{2}$ ≤ . . . ≤ x$_{m}$ = y . The homomorphism property gives Φ$_{i}$ ( x, y ) = φ ( x$_{1}$, x$_{2}$ ) · · · φ ( x$_{m}$$_{−}$$_{1}$, x$_{m}$ ) for i = 1 or i = 2. Thus Φ$_{1}$ = Φ$_{2}$ which proves σ is a G -essential set for ρ . Therefore σ is a G -grading set for ρ . We fix the notation δ ( ρ ) = { ( x, x ) : x ∈ X } for the diagonal subset of a relation ρ on a set X . The off-diagonal set ρ ∗ = ρ \ δ ( ρ ) is an anti-reflexive relation on X . Definition 4.1 Suppose ρ$_{1}$ and ρ$_{2}$ are relations on X$_{1}$ and X$_{2}$ , respectively. A function θ : X$_{2}$ → X$_{1}$ is called a compression map if 1, 2, and 3 are satisfied. In this case we say ρ$_{1}$ is a compression of ρ$_{2}$ . 1. θ is surjective and order-preserving. 2. For all a$_{1}$, a$_{2}$, a$_{3}$ ∈ X$_{1}$ if ( a$_{1}$, a$_{2}$, a$_{3}$ ) ∈ Trans ( X$_{1}$ ) then there exist x$_{1}$, x$_{2}$, x$_{3}$ ∈ X$_{2}$ such that ( x$_{1}$, x$_{2}$, x$_{3}$ ) ∈ Trans ( X$_{2}$ ) and θ ( x$_{i}$ ) = a$_{i}$ for i = 1 , 2 , 3 . 3. There is a bijection θ ∗ : ρ ∗ 2 → ρ ∗ 1 given by θ $^{∗}$( x, y ) = ( θ ( x ) , θ ( y )) for all x, y ∈ X$_{2}$ with xρ ∗ $_{2}$y . Compression maps were introduced in [8]. Example 5.3 uses a compression map to construct a G -grading set for a non-transitive relation. Lemma 4.2 Suppose I ( X$_{1}$, ρ$_{1}$, R ) and I ( X$_{2}$, ρ$_{2}$, R ) are generalized incidence rings and θ : X$_{2}$ → X$_{1}$ is a compression map. 11