Minimal and intrinsic topologies on monoids of elementary embeddings.

To every omega-categorical structure one can associate two spaces of symmetries which determine it up to first-order bi-interpretability: the topological group of its automorphisms and the topological monoid of its elementary embeddings, both equipped with the topology of pointwise convergence. We investigate when this topology is minimal amongst Hausdorff semigroup topologies on . The standard method to prove minimality of the topology of pointwise convergence for these monoids consists in showing that it coincides with the algebraically defined semigroup Zariski topology (Elliott, Jonušas, Mesyan, Mitchell, Morayne, and Péresse 2023; Pinsker and Schindler 2026). We show that that the semigroup Zariski topology on is not Hausdorff (and so cannot coincide with the topology of pointwise convergence) whenever has non-trivial centre. We then provide general conditions on the behaviour of algebraic closure in that imply minimality of the topology of pointwise convergence on . These conditions cover, for example, countable vector spaces and projective spaces over finite fields.

This is joint work with de la Nuez Gonzales, Ghadernezhad, and Pinsker.