Copies of the Random Graph: the 2-localization

Abstract

Let $G$ be a countable graph containing a copy of the countable random graph (Erdős-Rényi graph, Rado graph), $Emb(G)$ the monoid of its self-embeddings, $\mathbb{P}(G)=\{f[G]:f\in Emb(G)\}$ the set of copies of $G$ contained in $G$, and ${\mathcal{I}}_G$ the ideal of subsets of $G$ which do not contain a copy of $G$. We show that the poset $<\mathbb{P}(G),\subset >$, the algebra $P(G)/{\mathcal{I}}_G$, 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 $[b_{nm}:<n,m>\in \omega \times \omega ]$ of elements of $\mathbb{B}$ $${\bigwedge }_{n\in \omega }{\bigvee }_{m\in \omega }{b}_{nm}={\bigvee }_{\mathcal{T}\in Bt({}^{<\omega }\omega )}{\bigwedge }_{n\in \omega }{\bigvee }_{\phi \in \mathcal{T}\cap {}^{n+1}\omega }{\bigwedge }_{k\le n}{b}_{k\phi (k)},$$ where $Bt({}^{<\omega }\omega )$ denotes the set of all binary subtrees of the tree ${}^{<\omega }\omega$.

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