# Algebra Seminar talk

2024-01-19

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

Abstract:

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.