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Extensive categories, commutative semirings and Galois theory
We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2020
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| _version_ | 1867613246269161472 |
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| access_status_str | Open Access |
| author | Poklewski-Koziell, Rowan |
| author2 | Janelidze, George |
| author_browse | Janelidze, George Poklewski-Koziell, Rowan |
| author_facet | Janelidze, George Poklewski-Koziell, Rowan |
| author_sort | Poklewski-Koziell, Rowan |
| collection | Thesis |
| description | We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/32412 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:33:05.164Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/32412 Extensive categories, commutative semirings and Galois theory Poklewski-Koziell, Rowan Janelidze, George Applied Mathematics We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B. 2020-11-19T12:07:05Z 2020-11-19T12:07:05Z 2020 2020-11-19T08:42:00Z Master Thesis Masters MSc http://hdl.handle.net/11427/32412 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science |
| spellingShingle | Applied Mathematics Poklewski-Koziell, Rowan Extensive categories, commutative semirings and Galois theory |
| thesis_degree_str | Master's |
| title | Extensive categories, commutative semirings and Galois theory |
| title_full | Extensive categories, commutative semirings and Galois theory |
| title_fullStr | Extensive categories, commutative semirings and Galois theory |
| title_full_unstemmed | Extensive categories, commutative semirings and Galois theory |
| title_short | Extensive categories, commutative semirings and Galois theory |
| title_sort | extensive categories commutative semirings and galois theory |
| topic | Applied Mathematics |
| url | http://hdl.handle.net/11427/32412 |
| work_keys_str_mv | AT poklewskikoziellrowan extensivecategoriescommutativesemiringsandgaloistheory |