Supertropical semirings and supervaluations

Abstract

We interpret a valuation $\upsilon$ on a ring $R$ as a map $\upsilon:R \rightarrow M$ into a so called bipotent semiring $M$ (the usual max-plus setting), and then define a supervaluation $\varphi$ as a suitable map into a supertropical semiring $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $\upsilon$ via the ghost map $U \rightarrow M$. The set Cov($\upsilon$) of all supervaluations covering $\upsilon$ has a natural ordering which makes it a complete lattice. In the case that $R$ is a field, hence for $\upsilon$ a Krull valuation, we give a complete explicit description of Cov($\upsilon$). 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.