Supertropical semirings and supervaluations

We interpret a valuation v on a ring R as a map v:R→M into a so-called bipotent semiring M (the usual max–plus setting), and then define a supervaluationφ as a suitable map into a supertropical semiring U with ghost ideal M (cf. Izhakian and Rowen (2010, in press) [8,9]) covering v via the ghost map U→M. The set Cov(v) of all supervaluations covering v has a natural ordering which makes it a complete lattice. In the case where R is a field, and hence for v a Krull valuation, we give a completely explicit description of Cov(v).The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max–plus setting. We illustrate this by giving a supertropical version of Kapranov’s Lemma.