Valuations on semirings and supervaluations.
Abstract: Bourbaki´s notion of a valuation on a ring (Algebre commutative VI) readily generalizes to semirings,but ,of course, studying these leads to new phenomena.Such a valuation can best be viewed as a map into another semiring of special nature (a "bipotent" semiring).It is possible to lift this map to a map of special type with values in a so called "supertropical semiring" in various ways , which reveals a sort of complexity of the given valuation.(Joint work with Zur Ishakian and Louis Rowen.) .These are the " supervaluatios". They seem to deserve attention on their own in connection with tropical geometry. Valuations on semirings take an natural place in real algebra.The semiring of all sums of squares in a field is a case in point.