The Bergman property for endomorphism monoids of some Fra\"ıssé limits

Abstract

Based on an idea of Y. Péresse and some results of Maltcev, Mitchell and Ruškuc, 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.

16 pages; to appear in Forum Mathematicum

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• Universal homomorphisms, universal structures, and the polymorphism clones of homogeneous structures
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• Permutation monoids and MB-homogeneity for graphs and relational structures
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