On the lattice of weak topologies on the bicyclic monoid with adjoined zero
A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2020 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2020
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/188551 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the lattice of weak topologies on the bicyclic monoid with adjoined zero / S. Bardyla, O. Gutik // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 26–43. — Бібліогр.: 30 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862737775360475136 |
|---|---|
| author | Bardyla, S. Gutik, O. |
| author_facet | Bardyla, S. Gutik, O. |
| citation_txt | On the lattice of weak topologies on the bicyclic monoid with adjoined zero / S. Bardyla, O. Gutik // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 26–43. — Бібліогр.: 30 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-invariant filters on ! with an attached element 1 endowed with the following partial order: F ≤ G if and only if G = 1 or F ⊂ G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2ᶜ and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.
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| first_indexed | 2025-12-07T20:01:12Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-188551 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T20:01:12Z |
| publishDate | 2020 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Bardyla, S. Gutik, O. 2023-03-05T17:20:25Z 2023-03-05T17:20:25Z 2020 On the lattice of weak topologies on the bicyclic monoid with adjoined zero / S. Bardyla, O. Gutik // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 26–43. — Бібліогр.: 30 назв. — англ. 1726-3255 DOI:10.12958/adm1459 2010 MSC: 22A15, 06B23 https://nasplib.isofts.kiev.ua/handle/123456789/188551 A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-invariant filters on ! with an attached element 1 endowed with the following partial order: F ≤ G if and only if G = 1 or F ⊂ G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2ᶜ and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t. The work of the author is supported by the Austrian Science Fund FWF (grant I3709 N35). en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On the lattice of weak topologies on the bicyclic monoid with adjoined zero Article published earlier |
| spellingShingle | On the lattice of weak topologies on the bicyclic monoid with adjoined zero Bardyla, S. Gutik, O. |
| 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188551 |
| work_keys_str_mv | AT bardylas onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero AT gutiko onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero |