Beschreibung:
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 Phi and v on R, iming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation Phi : R - U is a multiplicative map from R to a supertropical U, cf. semiring [IR1], [IR2], [IKR1], with further properties, which mean that Phi 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 [B], while Phi >= v means that v:R - V is a sort of coarsening of the supervaluation Phi. If Phi(R) generates the semiring U, then Phi >= v if there exists a "transmission" Alpha:U - V with v=Alpha o Phi. Transmissions are multiplicative maps with further properties, cf. [IKR1, §55]. Every semiring homomorphism Alpha:U - 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.