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

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

Abstract

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

Original language	English
Pages (from-to)	1-30
Number of pages	30
Journal	Proceedings of the Edinburgh Mathematical Society
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
20M18
22A15
Baire space
poset of Polish topologies
symmetric inverse monoid
20M20
54H15

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

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AU - Mitchell, James

AU - Péresse, Yann

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

KW - Polish semigroup

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KW - 22A15

KW - Baire space

KW - poset of Polish topologies

KW - symmetric inverse monoid

KW - 20M20

KW - 54H15

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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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Keywords

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