2015/02/26 by Christian Pech, Pech, Christian, Maja Pech +1
Mathematics · #03C05 #03C10 #03C40 #08A35 #08A70 #08B25 #18A25 #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO) #Primary 03C15 #Rings and Algebras (math.RA) #Secondary 03C50 #math.CT #math.LO #math.RA #msc:03C05 #msc:03C10 #msc:03C15 #msc:03C40 #msc:03C50 #msc:08A35 #msc:08A70 #msc:08B25 #msc:18A25
paper · pdf · doi:10.48550/arxiv.1502.07769
38 pages, revised and extended version
arxiv created 2017/03/31 · arxiv updated 2017/04/04
Every clone of functions comes naturally equipped with a topology---the topology of pointwise convergence. A clone \mathfrakC is said to have automatic homeomorphicity with respect to a class C of clones, if every clone-isomorphism of \mathfrakC to a member of 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 ω-categorical structures. Our results extend and generalize previous results by Bodirsky, Pinsker, and Pongrácz.