On the lattice of weak topologies on the bicyclic monoid with adjoined zero
A Hausdorff topology \(\tau\) on the bicyclic monoid with adjoined zero \(\mathcal{C}^0\) is called weak if it is contained in the coarsest inverse semigroup topology on \(\mathcal{C}^0\). We show that the lattice \(\mathcal{W}\) of all weak shift-continuous topologies on \(\mathcal{C}^0\) is isomor...
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| Date: | 2020 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2020
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1459 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543238937116672 |
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| author | Bardyla, S. Gutik, O. |
| author_facet | Bardyla, S. Gutik, O. |
| author_sort | Bardyla, S. |
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| collection | OJS |
| datestamp_date | 2021-01-05T07:11:25Z |
| description | A Hausdorff topology \(\tau\) on the bicyclic monoid with adjoined zero \(\mathcal{C}^0\) is called weak if it is contained in the coarsest inverse semigroup topology on \(\mathcal{C}^0\). We show that the lattice \(\mathcal{W}\) of all weak shift-continuous topologies on \(\mathcal{C}^0\) is isomorphic to the lattice \(\mathcal{SIF}^1{\times}\mathcal{SIF}^1\) where \(\mathcal{SIF}^1\) is the set of all shift-invariant filters on \(\omega\) with an attached element \(1\) endowed with the following partial order: \(\mathcal{F}\leq \mathcal{G}\) if and only if \(\mathcal{G}=1\) or \(\mathcal{F}\subset \mathcal{G}\). Also, we investigate cardinal characteristics of the lattice \(\mathcal{W}\). In particular, we prove that \(\mathcal{W}\) contains an antichain of cardinality \(2^{\mathfrak{c}}\) and a well-ordered chain of cardinality \(\mathfrak{c}\). Moreover, there exists a well-ordered chain of first-countable weak topologies of order type \(\mathfrak{t}\). |
| first_indexed | 2025-12-02T15:30:07Z |
| format | Article |
| id | admjournalluguniveduua-article-1459 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:30:07Z |
| publishDate | 2020 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-14592021-01-05T07:11:25Z On the lattice of weak topologies on the bicyclic monoid with adjoined zero Bardyla, S. Gutik, O. lattice of topologies, bicyclic monoid, shift-continuous topology 22A15, 06B23 A Hausdorff topology \(\tau\) on the bicyclic monoid with adjoined zero \(\mathcal{C}^0\) is called weak if it is contained in the coarsest inverse semigroup topology on \(\mathcal{C}^0\). We show that the lattice \(\mathcal{W}\) of all weak shift-continuous topologies on \(\mathcal{C}^0\) is isomorphic to the lattice \(\mathcal{SIF}^1{\times}\mathcal{SIF}^1\) where \(\mathcal{SIF}^1\) is the set of all shift-invariant filters on \(\omega\) with an attached element \(1\) endowed with the following partial order: \(\mathcal{F}\leq \mathcal{G}\) if and only if \(\mathcal{G}=1\) or \(\mathcal{F}\subset \mathcal{G}\). Also, we investigate cardinal characteristics of the lattice \(\mathcal{W}\). In particular, we prove that \(\mathcal{W}\) contains an antichain of cardinality \(2^{\mathfrak{c}}\) and a well-ordered chain of cardinality \(\mathfrak{c}\). Moreover, there exists a well-ordered chain of first-countable weak topologies of order type \(\mathfrak{t}\). Lugansk National Taras Shevchenko University Austrian Science Fund FWF (Grant I 3709 N35) 2020-12-30 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1459 10.12958/adm1459 Algebra and Discrete Mathematics; Vol 30, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1459/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1459/585 Copyright (c) 2020 Algebra and Discrete Mathematics |
| spellingShingle | lattice of topologies bicyclic monoid shift-continuous topology 22A15 06B23 Bardyla, S. Gutik, O. On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title | On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_full | On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_fullStr | On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_full_unstemmed | On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_short | On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_sort | on the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| topic | lattice of topologies bicyclic monoid shift-continuous topology 22A15 06B23 |
| topic_facet | lattice of topologies bicyclic monoid shift-continuous topology 22A15 06B23 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1459 |
| work_keys_str_mv | AT bardylas onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero AT gutiko onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero |