Preservation of Equations by Monoidal Monads

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.