On semigroups which admit only discrete left-continuous Hausdorff topology
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910002550145024 |
|---|---|
| author | Gutik, Oleg |
| author_facet | Gutik, Oleg |
| contents | We give the sufficient condition when every left-continuous (right-continuous) Hausdorff topology on a semigroup $S$ is discrete. We construct a submonoid $\mathscr{C}_{+}(a,b)$ (resp., $\mathscr{C}_{-}(a,b)$) of the bicyclic monoid which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the following properties: $(i)$ every left-continuous (resp., right-continuous) Hausdorff topology on $S_α$ is discrete; $(ii)$ every semigroup $S_α$ admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); $(iii)$ every semigroup $S_α$ isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid $\mathscr{C}_{\mathbb{Z}}^+$ (resp., $\mathscr{C}_{\mathbb{Z}}^-$) of the extended bicyclic semigroup which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the above described properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19881 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On semigroups which admit only discrete left-continuous Hausdorff topology Gutik, Oleg Group Theory General Topology 22A15, 54C08, 54D10, 54D30, 54E52, 54H10 We give the sufficient condition when every left-continuous (right-continuous) Hausdorff topology on a semigroup $S$ is discrete. We construct a submonoid $\mathscr{C}_{+}(a,b)$ (resp., $\mathscr{C}_{-}(a,b)$) of the bicyclic monoid which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the following properties: $(i)$ every left-continuous (resp., right-continuous) Hausdorff topology on $S_α$ is discrete; $(ii)$ every semigroup $S_α$ admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); $(iii)$ every semigroup $S_α$ isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid $\mathscr{C}_{\mathbb{Z}}^+$ (resp., $\mathscr{C}_{\mathbb{Z}}^-$) of the extended bicyclic semigroup which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the above described properties. |
| title | On semigroups which admit only discrete left-continuous Hausdorff topology |
| topic | Group Theory General Topology 22A15, 54C08, 54D10, 54D30, 54E52, 54H10 |
| url | https://arxiv.org/abs/2601.19881 |