Copies of the Random Graph: the 2-localization

Abstract

Let $G$ be a countable graph containing a copy of the countable random graph (Erd\H{o}s-R\'enyi 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 $< Emb (G),\preceq ^R >$ 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 ${\mathbb B}$ $$\textstyle \bigwedge_{n \in \omega}\; \bigvee_{m \in \omega}\; b_{nm} = \bigvee_{{\mathcal T} \,\in \, Bt ({}^{<\omega}\omega)}\; \bigwedge_{n \in \omega}\; \bigvee_{\varphi \,\in \,{\mathcal T} \cap {}^{n+1}\omega}\; \bigwedge_{k\leq n}\; b_{k\varphi (k)}, $$ where $Bt ({}^{<\omega}\omega)$ denotes the set of all binary subtrees of the tree ${}^{<\omega}\omega$.