Valuations of semirings

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 (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 ${\mathbb{P}}_{{\mathbb{F}}_1}^1$ 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.