 Nondecidable exponential of decidable monoid sets

For the free monoid on $x,y$, act on $\mathbb{N}$ by adding word length and on $\{0,1\}$ trivially. Both actions are injective. The equivariant function $f(w,n)$ indicating words of length strictly greater than $n$ ending in $x$ is nonzero, but $y\cdot f=0$. Thus the exponential of monoid sets is not decidable. Strict inequality ensures invariance when the original word is empty.