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

TL;DR

This paper investigates the relationship between the Zariski and pointwise convergence topologies on endomorphism monoids of omega-categorical structures, establishing conditions for their equivalence and providing a counterexample.

Contribution

It identifies systematic conditions under which the Zariski and pointwise topologies coincide and presents the first known example where they differ, answering an open question.

Findings

The two topologies coincide under certain model-theoretic conditions.

The pointwise topology is the coarsest Hausdorff semigroup topology in these cases.

An explicit counterexample shows the topologies can differ on some 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. We establish two systematic reasons for the two topologies to agree, formulated in terms of the model-complete core of the structure. Further, we give an example of an $\omega$-categorical structure on whose endomorphism monoid the topology of pointwise convergence and the Zariski topology differ, answering a question of Elliott, Jonu\v{s}as, Mitchell, P\'eresse and Pinsker.