Valuations of semirings

Abstract

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 Qmax (the max-plus semifield of rational numbers) and also valuations on the ‘function field’ Qmax(T) (the semifield of rational functions over Qmax) which are trivial on Qmax. We construct and study the abstract curve associated to Qmax(T) in relation to the projective line PF11 over the field with one element F1 and the tropical projective line. Finally, we discuss possible connections to tropical curves and Berkovich's theory of analytic spaces.