The Bergman property for endomorphism monoids of some Fra\"{\i}ss\'e limits

TL;DR

This paper identifies conditions under which the endomorphism monoid of certain infinite ultrahomogeneous structures exhibits the Bergman property, linking permutation group theory and semigroup theory.

Contribution

It provides new sufficient conditions for the Bergman property in endomorphism monoids of countably infinite ultrahomogeneous structures.

Findings

Established a criterion for homomorphism-homogeneity.

Connected the Bergman property with ultrahomogeneous structures.

Extended previous results in permutation and semigroup theories.

Abstract

Based on an idea of Y. P\'eresse and some results of Maltcev, Mitchell and Ru\v{s}kuc, we present sufficient conditions under which the endomorphism monoid of a countably infinite ultrahomogeneous first-order structure has the Bergman property. This property has played a prominent role both in the theory of infinite permutation groups and, more recently, in semigroup theory. As a byproduct of our considerations, we establish a criterion for a countably infinite ultrahomogeneous structure to be homomorphism-homogeneous.