All countable monoids embed into the monoid of the infinite random graph
From the authors summary: ``We prove that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of \(\text{End}(R)\), the endomorphism monoid of the infinite random graph \(R\). Consequently, \(\text{End}(R)\) embeds each countable monoid, satisfies no nontrivial monoid identity, and has an undecidable universal theory.'' Note that the infinite random graph satisfies the existentially closed adjacency property, and it is a universal graph, i.e., each countable subgraph is isomorphic to an induced subgraph of \(R\).
scientific article; zbMATH DE number 5687567