Theorem 4.5 Assume ρ is a stable relation on a finite set X and every clasp in X is unlocked. 1. There is a preorder ≤ on a finite set Y and a compression map θ : Y → X . 2. There is an injective ring homomorphism h : I ( X, ρ, R ) → I ( Y, ≤ , R ) . 3. If G is a semigroup and I ( X, ρ, R ) has a grading induced by Φ : ρ → G then there is a good G -grading of I ( Y, ≤ , R ) such that h : I ( X, ρ, R ) → I ( Y, ≤ , R ) is an injective homomorphism of G -graded rings. Proof. Part 1 is [8, Theorem 3.5]. Part 2 is a special case of part 3 when G = { 1 } . Part 3 follows from part 1 and Lemma 4.2. 5 Group Gradings If ( X, ρ ) is a preorder then a subset A of X is a cross-cut if A is an antichain in X , for all x ∈ X there exists a ∈ A such that xρa or aρx , and if C is a chain in X then C can be extended to a chain C ′ such that A ∩ C ′ is nonempty. The length of a cross-cut A of X is | A | . For example, the minimal elements of X form a cross-cut. For a partially ordered with a cross-cut of length one or two there is a subset σ of ρ such that σ is a G -grading set for any group G (see [4, Theorem 4]). However if the shortest cross-cut of a partially ordered set has length three or more then there may not be a G -grading set for a group G (see [4, Example 6]). Theorem 5.1 Assume ( X, ρ ) is a preorder and X 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 any group G . 15