Some functor calculus on semirings
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916795734032384 |
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| author | Biswas, Pronay Goswami, Amartya Sardar, Sujit Kumar |
| author_facet | Biswas, Pronay Goswami, Amartya Sardar, Sujit Kumar |
| contents | We develop a functorial framework for the ideal theory of commutative semirings using coherent frames and spectral spaces. Two central constructions-the radical ideal functor and the $k$-radical ideal functor-are shown to yield coherent frames, with the latter forming a dense sublocale of the former. We define a natural transformation between these functors and analyze their categorical and topological properties.
Further, we introduce a notion of support that assigns to each semiring a bounded distributive lattice whose spectrum is homeomorphic to its prime spectrum. This enables the reconstruction of the radical ideal frame via lattice-theoretic data. Applications include adjunctions between quantales and complete idealic semirings; and a comparison of prime and $k$-prime spectra in semisimple $r$-semirings. Our results unify various spectral constructions in semiring theory through a categorical and pointfree perspective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some functor calculus on semirings Biswas, Pronay Goswami, Amartya Sardar, Sujit Kumar Rings and Algebras 16Y60, 06D22, 06D99, 06F07, 54B35, 06F30 We develop a functorial framework for the ideal theory of commutative semirings using coherent frames and spectral spaces. Two central constructions-the radical ideal functor and the $k$-radical ideal functor-are shown to yield coherent frames, with the latter forming a dense sublocale of the former. We define a natural transformation between these functors and analyze their categorical and topological properties. Further, we introduce a notion of support that assigns to each semiring a bounded distributive lattice whose spectrum is homeomorphic to its prime spectrum. This enables the reconstruction of the radical ideal frame via lattice-theoretic data. Applications include adjunctions between quantales and complete idealic semirings; and a comparison of prime and $k$-prime spectra in semisimple $r$-semirings. Our results unify various spectral constructions in semiring theory through a categorical and pointfree perspective. |
| title | Some functor calculus on semirings |
| topic | Rings and Algebras 16Y60, 06D22, 06D99, 06F07, 54B35, 06F30 |
| url | https://arxiv.org/abs/2506.13378 |