TY - JOUR

T1 - Supertropical semirings and supervaluations

AU - Izhakian, Zur

AU - Knebusch, Manfred

AU - Rowen, Louis

N1 - 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.

PY - 2011/10

Y1 - 2011/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79954629594&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2011.01.002

DO - 10.1016/j.jpaa.2011.01.002

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AN - SCOPUS:79954629594

SN - 0022-4049

VL - 215

SP - 2431

EP - 2463

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 10

ER -