Let $\mathcal M$ be a multiset and $\odot$ an associative operation over the support set $M$ of $\mathcal M$. Further, denote by $\mu_{\mathcal M}(x)$ the multiplicity of the element $x$ in $\mathcal M$. As usual in basic multiset theory, the predicate $x\in\mathcal M$ is defined to be true iff $\mu_{\mathcal M}(x)\geq1$.
My question is about the iterated application of the operation $\odot$ on (a function of) the elements of $\mathcal M$, where by "elements of $\mathcal M$" i mean each single occurrence of every element from the support $M$ as a distinct element in $\mathcal M$. Unfortunately, I was unable to find a notational convention for this purpose. More specifically, for any function $f:M\to M$, by writing $$\tag{1}\bigodot_{x\in\mathcal M}f(x)$$ by virtue of the predicate $\in$, I would expect this to mean $$\tag{2}\bigodot_{x\in M\,:\,\mu_{\mathcal M}(x)\geq1}f(x),$$ that is, the iterated application of $\odot$ on each element of $M$ which occurs at least once in $\mathcal M$, which is not what I described above. Rather, what I would like to express, in a compact notation, is the following: $$\tag{3}\bigodot_{x\in M}\bigodot_{i=1}^{\mu_{\mathcal M}(x)}f(x).$$
In conclusion, my question is the following: is there any convention addressing this issue? If yes, does $(1)$ correspond to $(2)$ or to $(3)$? Intuitively, I would say that the first holds, but I'd prefer the second option, or at least to have a less cumbersome notation for this case.
