Dominance and Transmissions in Supertropical Valuation Theory

Zur Izhakian, Manfred Knebusch, Louis Rowen

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9 Scopus citations

Abstract

This article is a sequel of [4], where we defined supervaluations on a commutative semiring R and studied a dominance relation φ{symbol} ≥ ψ between supervaluations φ{symbol} and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation φ{symbol}: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4], [7], [8], [5], [9], with further properties, which mean that φ{symbol} is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1], while φ{symbol} ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation φ{symbol}. If φ{symbol}(R) generates the semiring U, then φ{symbol} ≥ ψ iff there exists a "transmission" α: U → V with ψ = α ○ φ{symbol}. Transmissions are multiplicative maps with further properties, cf. [4, Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra.

Original languageEnglish
Pages (from-to)2736-2782
Number of pages47
JournalCommunications in Algebra
Volume41
Issue number7
DOIs
StatePublished - 2013
Externally publishedYes

Keywords

  • Bipotent semirings
  • Lattices
  • Monoid valuations
  • Supertropical algebra
  • Supertropical semirings
  • Supervaluations
  • Transmissive and homomorphic equivalence relations
  • Valuation theory

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