Metrizability of Clifford topological semigroups

Банах, Тарас; Гутік, Олег; Potiatynyk, Oles; Ravsky, Alex

doi:10.1007/s00233-011-9341-7

Semigroup Forum

2011

DOI: 10.1007/s00233-011-9341-7

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Metrizability of Clifford topological semigroups

Тарас Банах

Олег Гутік

Oles Potiatynyk

et al.

Abstract: Abstract. We prove that a topological Clifford semigroup S is metrizable if and only if S is an M -space and the set E = {e ∈ S : ee = e} of idempotents of S is a metrizable G δ -set in S. The same metrization criterion holds also for any countably compact Clifford topological semigroup S.

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2024

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Topological embeddings into transformation monoids

Bardyla,

Elliott,

Mitchell

et al. 2024

Forum Mathematicum

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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.

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Topological embeddings into transformation monoids

Bardyla,

Elliott,

Mitchell

et al. 2024

Forum Mathematicum

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show abstract

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Metrizability of Clifford topological semigroups

Cited by 1 publication

References 7 publications

Topological embeddings into transformation monoids

Topological embeddings into transformation monoids

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