Supertropical semirings and supervaluations

Abstract

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 de¯ne a supervaluation ' as a suitable map into a supertropical semiring U with ghost ideal M (cf. [IR1], [IR2]) 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 that R is a field, hence for v a Krull valuation, we give a complete 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.