A note on idempotent semirings
For a commutative semiring $S$, by an $S$-algebra we mean a commutative semiring $A$ equipped with a homomorphism $S\to A$. We show that the subvariety of $S$-algebras determined by the identities $1+2x=1$ and $x^2=x$ is closed under non-empty colimits. The (known) closedness of the category of Bool...
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| Format: | Article |
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Shahid Beheshti University
2025-01-01
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| Series: | Categories and General Algebraic Structures with Applications |
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| Online Access: | https://cgasa.sbu.ac.ir/article_104793_c3ad43e48e0ab598d02b1afd64b7dccb.pdf |
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| author | Manuela Sobral George Janelidze |
| author_facet | Manuela Sobral George Janelidze |
| author_sort | Manuela Sobral |
| collection | DOAJ |
| description | For a commutative semiring $S$, by an $S$-algebra we mean a commutative semiring $A$ equipped with a homomorphism $S\to A$. We show that the subvariety of $S$-algebras determined by the identities $1+2x=1$ and $x^2=x$ is closed under non-empty colimits. The (known) closedness of the category of Boolean rings and of the category of distributive lattices under non-empty colimits in the category of commutative semirings both follow from this general statement. |
| format | Article |
| id | doaj-art-247787f6330440958960fe1d9f86eb83 |
| institution | Kabale University |
| issn | 2345-5853 2345-5861 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Shahid Beheshti University |
| record_format | Article |
| series | Categories and General Algebraic Structures with Applications |
| spelling | doaj-art-247787f6330440958960fe1d9f86eb832025-01-24T18:43:40ZengShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532345-58612025-01-0122117518010.48308/cgasa.2024.235337.1484104793A note on idempotent semiringsManuela Sobral0George Janelidze1Department of Mathematics, Faculty of Science and Technology, University of Coimbra, Portugal.Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa.For a commutative semiring $S$, by an $S$-algebra we mean a commutative semiring $A$ equipped with a homomorphism $S\to A$. We show that the subvariety of $S$-algebras determined by the identities $1+2x=1$ and $x^2=x$ is closed under non-empty colimits. The (known) closedness of the category of Boolean rings and of the category of distributive lattices under non-empty colimits in the category of commutative semirings both follow from this general statement.https://cgasa.sbu.ac.ir/article_104793_c3ad43e48e0ab598d02b1afd64b7dccb.pdfcommutative semiringnon-empty colimitcoreflective subcategoryboolean algebradistributive lattice |
| spellingShingle | Manuela Sobral George Janelidze A note on idempotent semirings Categories and General Algebraic Structures with Applications commutative semiring non-empty colimit coreflective subcategory boolean algebra distributive lattice |
| title | A note on idempotent semirings |
| title_full | A note on idempotent semirings |
| title_fullStr | A note on idempotent semirings |
| title_full_unstemmed | A note on idempotent semirings |
| title_short | A note on idempotent semirings |
| title_sort | note on idempotent semirings |
| topic | commutative semiring non-empty colimit coreflective subcategory boolean algebra distributive lattice |
| url | https://cgasa.sbu.ac.ir/article_104793_c3ad43e48e0ab598d02b1afd64b7dccb.pdf |
| work_keys_str_mv | AT manuelasobral anoteonidempotentsemirings AT georgejanelidze anoteonidempotentsemirings AT manuelasobral noteonidempotentsemirings AT georgejanelidze noteonidempotentsemirings |