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

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)Emb(G) be the monoid of self-embeddings of G  , P(G)={f[G]:f∈Emb(G)}P(G)={f[G]:f∈Emb(G)} the set of copies of G contained in G  , and IGIG the ideal of subsets of G which do not contain a copy of G  . We show that the poset 〈P(G),⊂〉〈P(G),⊂〉, the algebra P(G)/IGP(G)/IG, and the inverse of the right Green's pre-order 〈Emb(G),⪯R〉〈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 [bnm:〈n,m〉∈ω×ω][bnm:〈n,m〉∈ω×ω] of elements of BB⋀n∈ω⋁m∈ωbnm=⋁T∈Bt(ω<ω)⋀n∈ω⋁φ∈T∩ωn+1⋀k≤nbkφ(k), where Bt(ω<ω) denotes the set of all binary subtrees of the tree ω<ω.