Polymorphism clones of homogeneous structures (Universal homogeneous polymorphisms and automatic homeomorphicity)

Christian Pech, Maja Pech

Published 2015-02-26Version 1

Every clone of functions comes naturally equipped with a topology---the topology of pointwise convergence. A clone $\mathfrak{C}$ is said to have automatic homeomorphicity with respect to a class $\mathcal{C}$ of clones, if every clone-isomorphism of $\mathfrak{C}$ to a member of $\mathcal{C}$ is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we study automatic homeomorphicity-properties for polymorphism clones of countable homogeneous relational structures. To this end we introduce and utilize universal homogeneous polymorphisms. Our results extend and generalize previous results by Bodirsky, Pinsker, and Pongr\'acz.