Polish topologies on endomorphism monoids of linear orders

Abstract

In this paper, we investigate Polish semigroup topologies on the endomorphism monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$. We introduce a new structural condition, property $\mathbb{XX}$, which yields automatic continuity of Borel measurable homomorphisms between certain topological semigroups. This provides a new method for analyzing Polish semigroup topologies on monoids with small groups of units. We show that for all monoids considered, the semigroup Zariski topology coincides with the pointwise topology and is therefore the coarsest Hausdorff semigroup topology. We prove that the submonoid $\operatorname{End}^{\infty}(\mathbb{N},\leq)$ of $\operatorname{End}(\mathbb{N},\leq)$ consisting of all endomorphisms with infinite image admits a unique Polish semigroup topology, namely the pointwise topology. On the other hand, despite possessing a finest Polish semigroup topology, the monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$, admit infinitely many distinct Polish semigroup topologies. Also, we show that the monoid $\operatorname{End}(\mathbb{N},<)$ admits exactly $2^{\aleph_0}$ Polish semigroup topologies and no maximal second-countable semigroup topology.