The endomorphism monoid of the random graph has uncountably many ideals. (Q1882669)

The paper concerns endomorphism monoids of graphs. The random graph \(R\) is the unique countable graph whose vertex set \(V(R)\) is the union of two sets \(P,Q\) such that \(P\cap Q=\emptyset\) and there exists a vertex \(u\in V(R)-(P\cup Q)\) adjacent to all vertices from \(P\) and to none from \(Q\). It is proved that its endomorphism monoid \(\text{End}(R)\) is not simple, but the lattice of the ideals of \(\text{End}(R)\) embeds the poset of all subsets of \(\omega\) (the set of all positive integers).