Polish topologies on endomorphism monoids of relational structures

Abstract

In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid $\mathrm{End}(\mathbb{A})$ 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.

21 pages

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