Automatic Homeomorphicity of Locally Moving Clones

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 $\mathrm{P}\mathrm{o}\mathrm{l}(M,\mathcal{L})$ is the polymorphism clone of a reduct of $(\mathbb{Q},<)$ or $(\mathbb{L},C)$ such that $$\mathrm{A}\mathrm{u}\mathrm{t}$(M,$\mathcal{L}$) \not= $\mathrm{A}\mathrm{u}\mathrm{t}$(M,=)$and$$\mathrm{E}\mathrm{n}\mathrm{d}$(M,$\mathcal{L}$) = $\mathrm{E}\mathrm{m}\mathrm{b}$(M,$\mathcal{L}$)$then$$\mathrm{P}\mathrm{o}\mathrm{l}$(M,$\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 Fra"{i}ss`{e} Generic Boolean algebra, then $\mathrm{P}\mathrm{o}\mathrm{l}(\mathcal{M})$ is locally moving.