All countable monoids embed in the monoid of the infinite random graph

Abstract. We prove that the endomorphism monoid of the infi-nite random graph R contains as a submonoid an isomorphic copy of each countable monoid. As a corollary, the monoid of R does not satisfy any non-trivial semigroup identity. We also prove that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of the monoid of R. 1