Supertropical semirings and supervaluations

Zur Izhakian; Manfred Knebusch; Louis Rowen

doi:10.1016/j.jpaa.2011.01.002

Supertropical semirings and supervaluations

Zur Izhakian, Manfred Knebusch, Louis Rowen

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

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
Pages (from-to)	2431-2463
Number of pages	33
Journal	Journal of Pure and Applied Algebra
Volume	215
Issue number	10
DOIs	https://doi.org/10.1016/j.jpaa.2011.01.002
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
Emmy Noether Research Institute for Mathematics
Gelbart Institute at Bar-Ilan University
Department of Mathematics, Bar-Ilan University
Israel Science Foundation	448/09
Tel Aviv University

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10.1016/j.jpaa.2011.01.002

Cite this

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title = "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 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.",

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Supertropical semirings and supervaluations

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