  If a monad $T$ is monoidal, then operations on a set $X$ can be lifted
canonically to operations on $TX$. In this paper we study structural properties
under which $T$ preserves equations between those operations. It has already
been shown that any monoidal monad preserves linear equations; affine monads
preserve drop equations (where some variable appears only on one side, such as
$x\cdot y = y$) and relevant monads preserve dup equations (where some variable
is duplicated, such as $x \cdot x = x$). We start the paper by showing a
converse: if the monad at hand preserves a drop equation, then it must be
affine. From this, we show that the problem whether a given (drop) equation is
preserved is undecidable. A converse for relevance turns out to be more subtle:
preservation of certain dup equations implies a weaker notion which we call
$n$-relevance. Finally, we identify the subclass of equations such that their
preservation is equivalent to relevance.
