Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups
A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup <i>G</i>, we offer a new way to construct a gyrogroup <inline-formula><math displa...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-11-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/11/1817 |
| _version_ | 1797549039088041984 |
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| author | Jaturon Wattanapan Watchareepan Atiponrat Teerapong Suksumran |
| author_facet | Jaturon Wattanapan Watchareepan Atiponrat Teerapong Suksumran |
| author_sort | Jaturon Wattanapan |
| collection | DOAJ |
| description | A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup <i>G</i>, we offer a new way to construct a gyrogroup <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula> such that <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula> contains a gyro-isomorphic copy of <i>G</i>. We then prove that every strongly topological gyrogroup <i>G</i> can be embedded as a closed subgyrogroup of the path-connected and locally path-connected topological gyrogroup <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula>. We also study several properties shared by <i>G</i> and <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula>, including gyrocommutativity, first countability and metrizability. As an application of these results, we prove that being a quasitopological gyrogroup is equivalent to being a strongly topological gyrogroup in the class of normed gyrogroups. |
| first_indexed | 2024-03-10T15:09:00Z |
| format | Article |
| id | doaj.art-5a80a77dd8394364b043580760fc19ba |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T15:09:00Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-5a80a77dd8394364b043580760fc19ba2023-11-20T19:31:11ZengMDPI AGSymmetry2073-89942020-11-011211181710.3390/sym12111817Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected GyrogroupsJaturon Wattanapan0Watchareepan Atiponrat1Teerapong Suksumran2Doctoral Program in Mathematics, Graduate School, Chiang Mai University, Chiang Mai 50200, ThailandDepartment of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, ThailandDepartment of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, ThailandA gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup <i>G</i>, we offer a new way to construct a gyrogroup <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula> such that <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula> contains a gyro-isomorphic copy of <i>G</i>. We then prove that every strongly topological gyrogroup <i>G</i> can be embedded as a closed subgyrogroup of the path-connected and locally path-connected topological gyrogroup <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula>. We also study several properties shared by <i>G</i> and <inline-formula><math display="inline"><semantics><msup><mi>G</mi><mo>•</mo></msup></semantics></math></inline-formula>, including gyrocommutativity, first countability and metrizability. As an application of these results, we prove that being a quasitopological gyrogroup is equivalent to being a strongly topological gyrogroup in the class of normed gyrogroups.https://www.mdpi.com/2073-8994/12/11/1817topological gyrogroupembedding of gyrogroupnormed gyrogroupgyrogroup extensionquasitopological gyrogroup |
| spellingShingle | Jaturon Wattanapan Watchareepan Atiponrat Teerapong Suksumran Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups Symmetry topological gyrogroup embedding of gyrogroup normed gyrogroup gyrogroup extension quasitopological gyrogroup |
| title | Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups |
| title_full | Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups |
| title_fullStr | Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups |
| title_full_unstemmed | Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups |
| title_short | Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups |
| title_sort | embedding of strongly topological gyrogroups in path connected and locally path connected gyrogroups |
| topic | topological gyrogroup embedding of gyrogroup normed gyrogroup gyrogroup extension quasitopological gyrogroup |
| url | https://www.mdpi.com/2073-8994/12/11/1817 |
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