Definition 4.4 Let ρ be a reflexive relation on a set X . Next we prove σ$_{1}$ is a G -extendible set for ρ$_{1}$ . Given ψ$_{1}$ : σ$_{1}$ → G we let ψ$_{2}$ : σ$_{2}$ → G be the function given by ψ$_{2}$ = ψ$_{1}$ ◦ θ $^{∗}$. There is also a homomorphism Ψ$_{2}$ : ρ$_{2}$ → G such that Ψ$_{2}$ |$_{σ}$$_{2}$ = ψ$_{2}$ since σ$_{2}$ is a G -extendible set for ρ$_{2}$ . There is a homomorphism Ψ$_{1}$ : ρ$_{1}$ → G such that for all a, b ∈ X$_{1}$ such that aρ$_{1}$b we have Ψ$_{1}$ ( a, b ) = Ψ$_{2}$ $^{(}$( θ $^{∗}$) − $^{1}$( a, b ) ) if a = b andΨ$_{1}$ ( a, b ) = 1 if a = b . Given a, b ∈ X$_{1}$ such that ( a, b ) ∈ σ$_{1}$ there exist x, y ∈ X$_{1}$ such that a = θ ( x ), b = θ ( y ), and ( x, y ) ∈ σ$_{2}$ . We have θ $^{∗}$( x, y ) = ( a, b ) and Ψ$_{2}$ ( x, y ) = Ψ$_{1}$ ( a, b ) by construction, Ψ$_{2}$ ( x, y ) = ψ$_{2}$ ( x, y ) since ( x, y ) ∈ σ$_{2}$ , and ψ$_{2}$ ( x, y ) = ψ$_{1}$ ( a, b ) since ψ$_{2}$ = ψ$_{1}$ ◦ θ $^{∗}$. This shows Ψ$_{1}$ |$_{σ}$$_{1}$ = ψ$_{1}$ and σ$_{1}$ is a G -extendible set for ρ$_{1}$ . 1. ( X, ρ ) is stable if ρ is balanced and if the relations aρb , aρc , bρc , bρd , and cρd imply aρd for all distinct a, b, c, d ∈ X . 2. An element x ∈ X is a clasp if there exist w, y ∈ X \ { x } such that wρx , xρy , and ( w, y ) / ∈ ρ . 3. x ∈ X is a locked clasp if there exist u, v, w, y ∈ X \ { x } such that ( w, y ) / ∈ ρ and ( u, x, y ) , ( u, x, v ) , ( w, x, v ) ∈ Trans ( X ) . 4. An unlocked clasp is a clasp which is not locked. It is easy to see a preorder is stable. The balanced relation determined by (d) in Figure 1 is not stable. Neither a balanced relation which is not stable nor a stable relation which contains a locked clasp can be the compression of a preorder by [8, Theorem 2.4 and Lemma 3.4]. 14