Abstract
AbstractUsing properties of unitary Cauchy filters on monothetic monoids, we prove a criterion of the existence of an embedding of such a monoid into a topological group. The proof of the sufficiency is constructive: under the corresponding assumptions, we are building a dense embedding of a given monothetic monoid into a monothetic group.
Highlights
The problem of embedding a topological semigroup into a topological group was investigated by a number of authors
Using properties of unitary Cauchy lters on monothetic monoids, we prove a criterion of the existence of an embedding of such a monoid into a topological group
The proof of the su ciency is constructive: under the corresponding assumptions, we are building a dense embedding of a given monothetic monoid into a monothetic group
Summary
The problem of embedding a topological semigroup into a topological group was investigated by a number of authors. MU = MU(F) and xU where the line on top denotes the topological closure in (Θ, τ|), U runs any base of τ| at the point , and x runs (for any U) an arbitrary set MU with U ≺ U This inequality means that there exists a neighborhood O of such that U O ⊂ U, and it implies that MU ⊂ MU. To Lemma 2.10 from [1], one can prove that it is generated by families of sets of the form MUs = MUs (F) and xU where U runs a base of neighborhoods of in Θ, and x runs, for any U, an arbitrary set MUs with U ≺s U This inequality means that there exists a neighborhood O of in Θ such that U O ⊂ U. For given U, let V be a neighborhood of in Θ such that pm−n ∈ U follows from pm , pn ∈ V and m≥n≥
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