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

Abstract

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. Next to two generic criteria for the automatic homeomorphicity of the polymorphism clones of free homogeneous structures we show that the polymorphism clone of the generic poset with reflexive ordering has automatic homeomorphicity and that the polymorphism clone of the generic poset with strict ordering has automatic homeomorphicity with respect to countable $\omega$-categorical structures. Our results extend and generalize previous results by Bodirsky, Pinsker, and Pongr\'acz.