Valuations of semirings
| Autor: | Jaiung Jun |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Rational number
Algebra and Number Theory 010102 general mathematics Rational function Field with one element Mathematics - Commutative Algebra Commutative Algebra (math.AC) 01 natural sciences Semiring Combinatorics Mathematics - Algebraic Geometry 14T99 13A18 Projective line 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Discrete valuation Algebraic Geometry (math.AG) Function field Semifield Mathematics |
| Zdroj: | Journal of Pure and Applied Algebra. 222:2063-2088 |
| ISSN: | 0022-4049 |
| Popis: | We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the implementation of hyperrings to yield a new definition (\emph{hyperfield valuation}). In particular, we classify valuations on the semifield $\mathbb{Q}_{max}$ (the max-plus semifield of rational numbers) and also valuations on the `function field' $\mathbb{Q}_{max}(T)$ (the semifield of rational functions over $\mathbb{Q}_{max}$) which are trivial on $\mathbb{Q}_{max}$. We construct and study the abstract curve associated to $\mathbb{Q}_{max}(T)$ in relation to the projective line $\mathbb{P}^1_{\mathbb{F}_1}$ over the field with one element $\mathbb{F}_{1}$ and the tropical projective line. Finally, we discuss possible connections to tropical curves and Berkovich's theory of analytic spaces. Comment: 24 pages, updated and extended, in the current version the tropical projective line is interpreted as an abstract curve in the semiring setting |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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