Dominance and Transmissions in Supertropical Valuation Theory

Abstract

This paper is a sequel of [IKR1], where we defined supervaluations on a commutative ring $R$ and studied a dominance relation $\Phi >= v$ between supervaluations $\varphi$ and $\upsilon$ on $R$, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation $\varphi : R \rightarrow U$ is a multiplicative map from $R$ to a supertropical semiring $U$, cf. [IR1], [IR2], [IKR1], with further properties, which mean that $\varphi$ is a sort of refinement, or covering, of an m-valuation (= monoid valuation) $\upsilon:R \rightarrow M$. In the most important case, that $R$ is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [B], while $\varphi >= \upsilon$ means that $\upsilon:R \rightarrow V$ is a sort of coarsening of the supervaluation $\varphi$. If $\varphi(R)$ generates the semiring $U$, then $\varphi >= \upsilon$ if there exists a "transmission" $\alpha:U \rightarrow V$ with $\varphi=\alpha \circ \varphi$. Transmissions are multiplicative maps with further properties, cf. [IKR1, §55]. Every semiring homomorphism $\alpha:U \rightarrow V$ is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the paper we study surjective transmissions via equivalence relations on supertropical semirings, often much more complicated than congruences by ideals in usual commutative algebra.