## 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 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.

Original language | English |
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Pages (from-to) | 2431-2463 |

Number of pages | 33 |

Journal | Journal of Pure and Applied Algebra |

Volume | 215 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2011 |

### Bibliographical note

Funding Information:This research of the first and third authors is supported by the Israel Science Foundation (grant No. 448/09). This research of the second author was supported in part by the Gelbart Institute at Bar-Ilan University, the Minerva Foundation at Tel-Aviv University, the Department of Mathematics of Bar-Ilan University, and the Emmy Noether Institute at Bar-Ilan University. This paper was completed under the auspices of the Research in Pairs program of the Mathematisches Forschungsinstitut Oberwolfach, Germany.

### Funding

This research of the first and third authors is supported by the Israel Science Foundation (grant No. 448/09). This research of the second author was supported in part by the Gelbart Institute at Bar-Ilan University, the Minerva Foundation at Tel-Aviv University, the Department of Mathematics of Bar-Ilan University, and the Emmy Noether Institute at Bar-Ilan University. This paper was completed under the auspices of the Research in Pairs program of the Mathematisches Forschungsinstitut Oberwolfach, Germany.

Funders | Funder number |
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Emmy Noether Institute at Bar-Ilan University | |

Gelbart Institute at Bar-Ilan University | |

Department of Mathematics, Bar-Ilan University | |

Israel Science Foundation | 448/09 |

Tel Aviv University |