Minimal and intrinsic topologies on monoids of elementary embeddings

To every $\omega$-categorical structure $M$ one can associate two spaces of symmetries which determine the structure up to first-order bi-interpretability: the topological group $\mathrm{A}\mathrm{u}\mathrm{t}(M)$ of its automorphisms and the topological monoid $\mathrm{E}\mathrm{E}\mathrm{m}\mathrm{b}(M)$ of its elementary embeddings, both equipped with the topology of pointwise convergence ${\tau }_{\mathrm{p}\mathrm{w}}$. We investigate the relation of ${\tau }_{\mathrm{p}\mathrm{w}}$ to other topologies on these spaces: in particular, when ${\tau }_{\mathrm{p}\mathrm{w}}$ is minimal, i.e.~does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of ${\tau }_{\mathrm{p}\mathrm{w}}$ on $\mathrm{E}\mathrm{E}\mathrm{m}\mathrm{b}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology ${\tau }_{\mathrm{Z}}$. We show that ${\tau }_{\mathrm{p}\mathrm{w}}$ differs from ${\tau }_{\mathrm{Z}}$ on $\mathrm{E}\mathrm{E}\mathrm{m}\mathrm{b}(M)$ whenever $\mathrm{A}\mathrm{u}\mathrm{t}(M)$ has non-trivial centre. We then provide general conditions on the behaviour of algebraic closure on $M$ that imply minimality of ${\tau }_{\mathrm{p}\mathrm{w}}$. These condition cover, for example, countable vector spaces and projective spaces over finite fields. Turning to $\mathrm{A}\mathrm{u}\mathrm{t}(M)$, we describe the minimal $T_1$ semigroup topologies on the automorphism groups of model-theoretically simple one-based $\omega$-categorical structures with weak elimination of imaginaries. We conclude by proving that the metric pointwise topology ${\tau }_{\mathrm{m}\mathrm{p}\mathrm{w}}$ is minimal, equals ${\tau }_{\mathrm{Z}}$, and is strictly coarser than ${\tau }_{\mathrm{p}\mathrm{w}}$, on $\mathrm{E}\mathrm{E}\mathrm{m}\mathrm{b}(M)$ for the real and the rational Urysohn space and sphere.