Topological embeddings into transformation monoids

Serhii Bardyla*, Luke Elliott, James D. Mitchell, Yann Péresse

*Corresponding author for this work

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In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid ℕ or the symmetric inverse monoid I with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into ℕ and belong to any of the following classes: commutative semigroups, compact semigroups, groups, and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and I. We construct several examples of countable Polish topological semigroups that do not embed into ℕ, which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of ℕ. The former complements recent works of Banakh et al.
Original languageEnglish
Number of pages18
JournalForum Mathematicum
Early online date6 Jan 2024
Publication statusE-pub ahead of print - 6 Jan 2024


  • Transformation monoid
  • Baire space
  • Polish semigroup
  • Topological embedding
  • Clifford semigroup


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