Polymorphism clones of homogeneous structures: gate coverings and automatic homeomorphicity

Abstract

© 2018, Springer International Publishing AG, part of Springer Nature. Every clone of functions comes naturally equipped with a topology, the topology of pointwise convergence. A clone C is said to have automatic homeomorphicity with respect to a class K of clones, if every clone isomorphism of C to a member of K 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. Besides two generic criteria for the automatic homeomorphicity of the polymorphism clones of homogeneous structures we show that the polymorphism clone of the generic poset with strict ordering has automatic homeomorphicity with respect to the class of polymorphism clones of countable ω-categorical structures. Our results extend and generalize previous results by Bodirsky, Pinsker, and Pongrácz.