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}
}
Supertropical Quadratic Forms I
Supertropical Monoids: Basics, Canonical Factorization, and Lifting Ghosts to Tangibles
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
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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Commutative ν-Algebra and Supertropical Algebraic Geometry
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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
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
Supertropical Quadratic Forms II
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
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References

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Dominance and Transmissions in Supertropical Valuation Theory
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
Supertropical matrix algebra
AbstractThe objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: The tropical determinant (i.e.,
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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
Tropical Mathematics
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
Supertropical matrix algebra II: Solving tropical equations
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
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