Algebra Seminar talk

Clemens Schindler
On the Zariski topology on endomorphism monoids of omega-categorical structures

The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the other hand, the Zariski topology induced within the monoid by (non-)solutions to equations. For all concrete endomorphism monoids of omega-categorical structures on which the Zariski topology has been analysed thus far, the two topologies were shown to coincide, in turn yielding that the pointwise topology is the coarsest Hausdorff semigroup topology on those endomorphism monoids. I will present two systematic reasons for the two topologies to agree, formulated in terms of the model-complete core of the structure. Further, I will give an explicit example of an omega-categorical structure, satisfying various model-theoretic wellbehavedness properties yet failing the condition on the core, on whose endomorphism monoid the topology of pointwise convergence and the Zariski topology differ, answering a question of Elliott, Jonušas, Mitchell, Péresse and Pinsker.

This is joint work with Michael Pinsker.