Topological embeddings into transformation monoids

Abstract

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