On algebraic congruence varieties over semirings
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2015
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| _version_ | 1866909435393212416 |
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| author | Qiu, Derong |
| author_facet | Qiu, Derong |
| contents | In this paper, we develop some foundations for a theory of algebraic varieties of congruences on commutative semirings. By studying the structure of congruences, firstly, we show that the spectrum $ \text{Spec}^{c}(A) $ consisting of prime congruences on a semirings $ A $ has a Zariski topological structure; Then, for two semirings $ A \subset B, $ we consider the polynomial semiring $ S = A[x_{1}, \cdots , x_{n}] $ and the affine $ n-$space $ B^{n}. $ For any congruence $ σ$ on $ S $ and congruence $ ρ$ on $ B, $ we introduce the $ ρ-$algebraic varieties $ Z_{ρ}(σ)(B)$ in $ B^{n}, $ which are the set of zeros in $ B^{n} $ of the system of polynomial $ ρ-$congruence equations given by $ σ. $ When $ ρ$ is a prime congruence, we find these varieties satisfying the axiom of closed sets, and forming a (Zariski) topology on $ B^{n}. $ Some results about their structures including a version of Nullstellensatz of congruences are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1512_08088 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | On algebraic congruence varieties over semirings Qiu, Derong Rings and Algebras Algebraic Geometry 14A10, 16Y60, 08A30, 14A05, 14A25, 14T05, 12E12 In this paper, we develop some foundations for a theory of algebraic varieties of congruences on commutative semirings. By studying the structure of congruences, firstly, we show that the spectrum $ \text{Spec}^{c}(A) $ consisting of prime congruences on a semirings $ A $ has a Zariski topological structure; Then, for two semirings $ A \subset B, $ we consider the polynomial semiring $ S = A[x_{1}, \cdots , x_{n}] $ and the affine $ n-$space $ B^{n}. $ For any congruence $ σ$ on $ S $ and congruence $ ρ$ on $ B, $ we introduce the $ ρ-$algebraic varieties $ Z_{ρ}(σ)(B)$ in $ B^{n}, $ which are the set of zeros in $ B^{n} $ of the system of polynomial $ ρ-$congruence equations given by $ σ. $ When $ ρ$ is a prime congruence, we find these varieties satisfying the axiom of closed sets, and forming a (Zariski) topology on $ B^{n}. $ Some results about their structures including a version of Nullstellensatz of congruences are obtained. |
| title | On algebraic congruence varieties over semirings |
| topic | Rings and Algebras Algebraic Geometry 14A10, 16Y60, 08A30, 14A05, 14A25, 14T05, 12E12 |
| url | https://arxiv.org/abs/1512.08088 |