Classifying the Polish semigroup topologies on the symmetric inverse monoid

Serhii Bardyla; Luna Elliott; James Mitchell; Yann Péresse

doi:10.1017/s0013091525101302

Classifying the Polish semigroup topologies on the symmetric inverse monoid

Serhii Bardyla^*, Luna Elliott, James Mitchell, Yann Péresse

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We classify all Polish semigroup topologies on the symmetric inverse monoid I_ℕ on the natural numbers 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_ℕ endowed with any second countable T₁ semigroup topology is homeomorphic to the Baire space ℕ^ℕ .

Original language	English
Pages (from-to)	1-30
Number of pages	30
Journal	Proceedings of the Edinburgh Mathematical Society
Volume	FirstView
Early online date	6 Mar 2026
DOIs	https://doi.org/10.1017/s0013091525101302
Publication status	E-pub ahead of print - 6 Mar 2026

Keywords

Polish semigroup
Baire space
Poset of Polish topologies
Symmetric inverse monoid

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10.1017/s0013091525101302Licence: CC BY

Bardyla-et-al-2026-Classifying-the-Polish-semigroup-ProcEdinbMathSoc-CCBY
Copyright © The Author(s), 2026. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Final published version, 606 KBLicence: CC BY

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@article{816637b25f9e49dea6fc7985331794e6,

title = "Classifying the Polish semigroup topologies on the symmetric inverse monoid",

abstract = "We classify all Polish semigroup topologies on the symmetric inverse monoid Iℕ on the natural numbers 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 Iℕ endowed with any second countable T1 semigroup topology is homeomorphic to the Baire space ℕℕ .",

keywords = "Polish semigroup, Baire space, Poset of Polish topologies, Symmetric inverse monoid",

author = "Serhii Bardyla and Luna Elliott and James Mitchell and Yann P{\'e}resse",

note = "Funding: The research of the first named author was funded in whole by the Austrian Science Fund FWF [10.55776/ESP399].",

year = "2026",

month = mar,

day = "6",

doi = "10.1017/s0013091525101302",

language = "English",

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T1 - Classifying the Polish semigroup topologies on the symmetric inverse monoid

AU - Bardyla, Serhii

AU - Elliott, Luna

AU - Mitchell, James

AU - Péresse, Yann

N1 - Funding: The research of the first named author was funded in whole by the Austrian Science Fund FWF [10.55776/ESP399].

PY - 2026/3/6

Y1 - 2026/3/6

N2 - We classify all Polish semigroup topologies on the symmetric inverse monoid Iℕ on the natural numbers 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 Iℕ endowed with any second countable T1 semigroup topology is homeomorphic to the Baire space ℕℕ .

AB - We classify all Polish semigroup topologies on the symmetric inverse monoid Iℕ on the natural numbers 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 Iℕ endowed with any second countable T1 semigroup topology is homeomorphic to the Baire space ℕℕ .

KW - Polish semigroup

KW - Baire space

KW - Poset of Polish topologies

KW - Symmetric inverse monoid

U2 - 10.1017/s0013091525101302

DO - 10.1017/s0013091525101302

M3 - Article

SN - 0013-0915

VL - FirstView

SP - 1

EP - 30

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

ER -

Classifying the Polish semigroup topologies on the symmetric inverse monoid

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