Classifying the Polish semigroup topologies on the symmetric inverse monoid
Abstract
Abstract We classify all Polish semigroup topologies on the symmetric inverse monoid $I_{\mathbb N}$ on the natural numbers $\mathbb N$ . This result answers a question of Elliott et al. There are countably infinitely many such topologies. Under containment, these Polish semigroup topologies form a join-semilattice with infinite descending chains, no infinite ascending chains, and arbitrarily large finite anti-chains. Also, we show that the monoid $I_{\mathbb N}$ endowed with any second countable $T_1$ semigroup topology is homeomorphic to the Baire space $\mathbb N^{\mathbb N}$ .
Fields of Study & Subjects
semigroups and automata theory
Functional Equations Stability Results
Publication details
- Journal / source
- Proceedings of the Edinburgh Mathematical Society
- Publisher
- Cambridge University Press (CUP)
- Pages
- 1–30
- Publication date
- 6 March 2026
- Language
- EN