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 $\mathrm{Pol}(M,\mathcal{L})$ is the polymorphism clone of a reduct of $(\mathbb{Q},<)$ or $(\mathbb{L},C)$ such that $\mathrm{Aut}(M,\mathcal{L}) \not= \mathrm{Aut}(M,=)$ 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 Fra\"{i}ss\`{e} Generic Boolean algebra, then $\mathrm{Pol}(\mathcal{M})$ is locally moving.