Topological embeddings into transformation monoids

Abstract

In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid ${\mathbb{N}}^{\mathbb{N}}$ or the symmetric inverse monoid $I_{\mathbb{N}}$ with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into ${\mathbb{N}}^{\mathbb{N}}$ 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 $I_{\mathbb{N}}$. We construct several examples of countable Polish topological semigroups that do not embed into ${\mathbb{N}}^{\mathbb{N}}$, 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 ${\mathbb{N}}^{\mathbb{N}}$. The former complements recent works of Banakh et al.

arXiv comment19 pages, update according to referees comments

References in corpus (10)

• Generating self-map monoids of infinite sets
• Monoids of injective maps closed under conjugation by permutations
• On automatic homeomorphicity for transformation monoids
• Polish topologies on endomorphism monoids of relational structures
• The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups
• A topological characterisation of endomorphism monoids of countable structures
• Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones
• The Bergman-Shelah Preorder on Transformation Semigroups
• Metrizability of Clifford topological semigroups
• On topological Clifford semigroups embeddable into products of cones over topological groups

Cited by in corpus (1)

• Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones