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

中文翻译：

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变换幺半群的拓扑嵌入

在本文中，我们考虑哪些拓扑半群拓扑嵌入到全变换幺半群中的问题 ℕ ℕ {\mathbb{N}^{\mathbb{N}}} 或对称反幺半群 我 ℕ {I_{\mathbb{N}}} 及其各自的规范波兰半群拓扑。我们描述那些拓扑嵌入的拓扑半群 ℕ ℕ {\mathbb{N}^{\mathbb{N}}} 且属于以下任意类别：交换半群、紧半群、群和某些 Clifford 半群。我们证明了拓扑逆半群的类似特征 我 ℕ {I_{\mathbb{N}}} 。我们构造了几个可数波兰拓扑半群的例子，它们不嵌入 ℕ ℕ {\mathbb{N}^{\mathbb{N}}} ，这对 Elliott 等人最近提出的一个开放问题给出了否定的回答。此外，我们获得了拓扑 Clifford 半群可度量的两个充分条件，并证明反演在每个 Clifford 子半群中自动连续 ℕ ℕ {\mathbb{N}^{\mathbb{N}}} 。前者补充了巴纳赫等人最近的作品。