2023/02/17 by S. Bardyla, Bardyla, S., L. Elliott +5
Computer Science · Mathematics · #20M18 #20M20 #20M30 #54E35 #54H15 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2302.08988
openalex publication_date 2023/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 Iℕ 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 Iℕ. 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.