# Supertropical semirings and supervaluations

@article{Izhakian2010SupertropicalSA,
title={Supertropical semirings and supervaluations},
author={Zur Izhakian and Manfred Knebusch and Louis H. Rowen},
journal={arXiv: Commutative Algebra},
year={2010}
}
• Published 4 March 2010
• Mathematics
• arXiv: Commutative Algebra
42 Citations
Supertropical Monoids: Basics, Canonical Factorization, and Lifting Ghosts to Tangibles
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• 2011
Supertropical monoids are a structure slightly more general than the supertropical semirings, which have been introduced and used by the first and the third authors for refinements of tropical
Dominance and Transmissions in Supertropical Valuation Theory
• Mathematics
• 2011
This article is a sequel of [4], where we defined supervaluations on a commutative semiring R and studied a dominance relation ϕ ≥ ψ between supervaluations ϕ and ψ on R, aiming at an enrichment of
Valuations of semirings
• Jaiung Jun
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Journal of Pure and Applied Algebra
• 2018
Commutative ν-Algebra and Supertropical Algebraic Geometry
This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over
Scheme theoretic tropicalization
In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a
Basic Operations on Supertropical Quadratic Forms
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• 2017
In the case that a module V over a (commutative) supertropical semiring R is free, the R-module Quad(V ) of all quadratic forms on V is almost never a free module. Nevertheless, Quad(V ) has two free
• Mathematics
• 2015
This article is a sequel of [4], where we introduced quadratic forms on a module V over a supertropical semiring R and analysed the set of bilinear companions of a quadratic form q : V → R in case

## References

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Dominance and Transmissions in Supertropical Valuation Theory
• Mathematics
• 2011
This article is a sequel of [4], where we defined supervaluations on a commutative semiring R and studied a dominance relation ϕ ≥ ψ between supervaluations ϕ and ψ on R, aiming at an enrichment of
THE M-VALUATION SPECTRUM OF A COMMUTATIVE RING
Harrison and Vitulli introduced the notion of a V-valuation of a commutative ring, which includes the standard concepts of valuation due to Krull, Artin, Samuel and Manis (see [HV] and the
On valuation spectra
If K is an ordered field then every convex subring of K is a valuation ring of K. This easy but fundamental observation has made valuation theory a very natural and important tool in real algebraic
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Abstract We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For
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We analyse the interplay between real valuations, Prufer extensions and convexity with respect to various preorderings on a given commutative ring. We study all this first in preordered rings in
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This article is based on the Clay Mathematics Senior Scholar Lecture that was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture was the tropical approach in
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We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of A, which provides the unique right (resp. left) quasi-inverse maximal with respect to