  We first develop a general theory of Johnson filtrations and Johnson
homomorphisms for a group $G$ acting on another group $K$ equipped with a
filtration indexed by a "good" ordered commutative monoid. Then, specializing
it to the case where the monoid is the additive monoid $\mathbb{N}^2$ of pairs
on nonnegative integers, we obtain a theory of double Johnson filtrations and
homomorphisms. We apply this theory to the mapping class group $\mathcal{M}$ of
a surface $\Sigma_{g,1}$ with one boundary component, equipped with the normal
subgroups $\bar{X}$, $\bar{Y}$ of $\pi_1(\Sigma_{g,1})$ associated to a
standard Heegaard splitting of the $3$-sphere. We also consider the case where
the group $G$ is the automorphism group of a free group.
