Topological embeddings into transformation monoids

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