Abstract
We interpret a valuation nu on a ring R as a map nu : R -> M into a so-called bipotent semiring M (the usual max-plus setting), and then define a supervaluation phi as a suitable map into a supertropical semiring U with ghost ideal M (cf. Izhakian and Rowen (2010, in press) [8,9]) covering nu via the ghost map U -> M. The set Cov(nu) of all supervaluations covering nu has a natural ordering which ...
Abstract
We interpret a valuation nu on a ring R as a map nu : R -> M into a so-called bipotent semiring M (the usual max-plus setting), and then define a supervaluation phi as a suitable map into a supertropical semiring U with ghost ideal M (cf. Izhakian and Rowen (2010, in press) [8,9]) covering nu via the ghost map U -> M. The set Cov(nu) of all supervaluations covering nu has a natural ordering which makes it a complete lattice. In the case where R is a field, and hence for a nu Krull valuation, we give a completely explicit description of Cov(nu). 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. (C) 2011 Elsevier B.V. All rights reserved.
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