  This paper is a sequel of [IKR1], where we defined supervaluations on a
commutative ring $R$ and studied a dominance relation $\phi \geq \psi$ between
supervaluations $\phi$ and $\psi$ on $R$, aiming at an enrichment of the
algebraic tool box for use in tropical geometry.
  A supervaluation $\phi:R \to U$ is a multiplicative map from $R$ to a
supertropical semiring $U$, cf. [IR1], [IR2], [IKR1], with further properties,
which mean that $\phi$ is a sort of refinement, or covering, of an m-valuation
(= monoid valuation) $v: R \to M$. 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 $\phi \geq \psi$ means that $\psi: R \to V$ is a sort of
coarsening of the supervaluation $\phi$. If $\phi(R)$ generates the semiring
$U$, then $\phi \geq \psi$ iff there exists a "transmission" $\alpha: U \to V$
with $\psi = \alpha \circ \phi$.
  Transmissions are multiplicative maps with further properties, cf. [IKR1,
Sec. 5]. Every semiring homomorphism $\alpha: U \to V$ 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.
