Polish topologies on endomorphism monoids of relational structures

TL;DR

This paper investigates the topological properties of endomorphism monoids of certain countable relational structures, establishing uniqueness of their Polish semigroup topologies and exploring automatic continuity.

Contribution

It introduces techniques to characterize minimal and maximal semigroup topologies and proves the uniqueness of Polish topologies for several well-known structures.

Findings

Endomorphism monoids of structures like the random graph have unique Polish semigroup topologies.

These topologies are subspace topologies of the Baire space.

Many structures exhibit automatic continuity for homomorphisms to second countable semigroups.

Abstract

In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid $\operatorname{End}(\mathbb{A})$ of a countable relational structure $\mathbb{A}$. As applications, we show that the endomorphism monoids of several well-known relational structures, including the random graph, the random directed graph, and the random partial order, possess a unique Polish semigroup topology. In every case this unique topology is the subspace topology induced by the usual topology on the Baire space $\mathbb{N} ^ \mathbb{N}$. We also show that many of these structures have the property that every homomorphism from their endomorphism monoid to a second countable topological semigroup is continuous; referred to as automatic continuity. Many of the results about endomorphism monoids are extended to clones of polymorphisms on the same structures.