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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| Veröffentlicht in: | Algebra and Discrete Mathematics |
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут прикладної математики і механіки НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/188551 |
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| Zitieren: | 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. |
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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 |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| spellingShingle |
On the lattice of weak topologies on the bicyclic monoid with adjoined zero Bardyla, S. Gutik, O. |
| title_short |
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_sort |
on the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| author |
Bardyla, S. Gutik, O. |
| author_facet |
Bardyla, S. Gutik, O. |
| publishDate |
2020 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| 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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| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/188551 |
| 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. |
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AT bardylas onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero AT gutiko onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero |
| first_indexed |
2025-12-07T20:01:12Z |
| last_indexed |
2025-12-07T20:01:12Z |
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1850880999666221056 |