Positive answers to Koch’s problem in special cases
| Autor: | Alex Ravsky, Taras Banakh, Oleg Gutik, Igor Guran, Serhii Bardyla |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Monoid
Mathematics::General Topology Topological semigroup cancellative semigroup Combinatorics QA1-939 non-viscous monoid 22a15 Locally compact space Topological group Mathematics Algebra and Number Theory feebly compact semigroup Topological monoid Group (mathematics) Applied Mathematics monothetic semigroup 54d30 tkachenko-tomita group countably compact semigroup Cancellative semigroup locally compact semigroup topological semigroup semitopo-logical semigroup Geometry and Topology koch’s problem Element (category theory) |
| Zdroj: | Topological Algebra and its Applications, Vol 8, Iss 1, Pp 76-87 (2020) |
| ISSN: | 2299-3231 |
| DOI: | 10.1515/taa-2020-0007 |
| Popis: | A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1. |
| Databáze: | OpenAIRE |
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