A universality result for endomorphism monoids of some ultrahomogeneous structures

TL;DR

This paper establishes a general condition under which the endomorphism monoid of certain ultrahomogeneous structures can embed all countable semigroups, unifying previous results and enabling new applications.

Contribution

It introduces a broad sufficient condition for embedding all countable semigroups into endomorphism monoids of ultrahomogeneous structures, unifying and extending prior work.

Findings

Provides a framework unifying previous scattered results.

Yields new applications for the Urysohn space and ultrahomogeneous semilattice.

Demonstrates the embedding of all countable semigroups into specific endomorphism monoids.

Abstract

We devise a fairly general sufficient condition ensuring that the endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a Fra\"{\i}ss\'{e} limit) embeds all countable semigroups. This approach provides us not only with a framework unifying the previous scattered results in this vein, but actually yields new applications for endomorphism monoids of the (rational) Urysohn space and the countable universal ultrahomogeneous semilattice.