Automatic Homeomorphicity of Locally Moving Clones

Abstract

We extend the work of M. Rubin on locally moving groups to clones, showing that a locally moving polymorphism clone has automatic homeomorphicity with respect to the class of all polymorphism clones. We show that if $\mathrm{Pol}(M,\mathcal{L})$ is the polymorphism clone of a reduct of $(\mathbb{Q},<)$ or $(\mathbb{L},C)$ such that $\mathrm{Aut}(M,\mathcal{L})\ne \mathrm{Aut}(M,=)$ and $\mathrm{End}(M,\mathcal{L})=\mathrm{Emb}(M,\mathcal{L})$ then $\mathrm{Pol}(M,\mathcal{L})$ is locally moving (and hence has automatic homeomorphicity with respect to the class of all polymorphism clones), where $\mathbb{Q}$ is the rationals, $(\mathbb{L},C)$ is the infinite binary-branching homogeneous $C$-relation. We also show that if $\mathcal{M}=(\mathbb{B},\cup ,\cap ,{}^c,1,0)$, the Fraïssè Generic Boolean algebra, then $\mathrm{Pol}(\mathcal{M})$ is locally moving.

References in corpus (2)

• Locally Moving Groups and the Reconstruction Problem for Chains and Circles
• Reconstructing the topology on monoids and polymorphism clones of the rationals

Cited by in corpus (3)

• Reconstructing the topology on monoids and polymorphism clones of reducts of the rationals
• Reconstruction theorems for semigroups of functions which contain all transpositions of a set and for clones with the same property
• Reconstructing the topology of the elementary self-embedding monoids of countable saturated structures