The poset of all copies of the random graph has the 2-localization property

Abstract Let G be a countable graph containing a copy of the countable universal and homogeneous graph, also known as the random graph. Let Emb ( G ) be the monoid of self-embeddings of G, P ( G ) = { f [ G ] : f ∈ Emb ( G ) } the set of copies of G contained in G, and I G the ideal of subsets of G which do not contain a copy of G. We show that the poset 〈 P ( G ) , ⊂ 〉 , the algebra P ( G ) / I G , and the inverse of the right Green's pre-order 〈 Emb ( G ) , ⪯ R 〉 have the 2-localization property. The Boolean completions of these pre-orders are isomorphic and satisfy the following law: for each double sequence [ b n m : 〈 n , m 〉 ∈ ω × ω ] of elements of B ⋀ n ∈ ω ⋁ m ∈ ω b n m = ⋁ T ∈ Bt ( ω ω ) ⋀ n ∈ ω ⋁ φ ∈ T ∩ ω n + 1 ⋀ k ≤ n b k φ ( k ) , where Bt ( ω ω ) denotes the set of all binary subtrees of the tree ω ω .