How to define a map from a multiset to a multiset

Question

Multisets are like sets, but can contain duplicates. Suppose we have two multisets $A, B$, and we want to define a map from one to the other ie:

$$ f : A \rightarrow B.$$

How do we do this? Is it possible to assign one "element" of the multiset $B$ with on element of the multiset $A$? Does it behave like a function, in that for every "element" of $A$, there is exactly one corresponding "element" of $B$? I think this must be equal to the span:

$$Set(A) \leftarrow C \rightarrow Set(B)$$

for some indexing set $C$, where $Set(A)$ is the underlying set of the multiset $A$. Is that true?

Answer 1

Suppose $A$ is a multiset where $a\in A$ appears more than once, i.e. $a_1,a_2\in A$. If $a_1$ and $a_2$ are indistinguishable, then there is no way to have $f(\{a_1\}) \neq f(\{a_2\})$. Similarly, if $b_1,b_2\in B$ are indistinguishable, then there is no way to have $f^{-1}(\{b_1\}) \neq f^{-1}(\{b_2\})$. Thus, any map between $A$ and $B$ will actually be a map between their underlying sets.

Answer 2

When considering a multiset $\{ a, b, a, a, b \}$, there is always two notions of "same": the finest one allowing to see that the first $a$ is not the "same" as the second $a$, and a coarser one declaring that are the same. This can be modeled in different ways, for example a set $X = \{(a,1), (a,2), (a,3), (b,1), (b,2)\}$ and an equivalence relation $(x,i) \equiv (x,j)$. Choosing to work with the coarsest "same" drops the multiset structure completely, so one usually works with the finest one, possibly with additional properties allowing to take into account the multiplicity. The above point of view in mentionned here, with other ones.

That being said, in almost every area of mathematics, one needs to choose both its objects, and the relevant notion of maps between the objects for its purpose. For example, having two sets $X$ and $Y$ does not say if one should use functions $f : X \to Y$, relations $R \subseteq X\times Y$, partial functions $g : X \rightharpoonup Y$ , etc. Your idea of a span is closer to a relation, a kind of "multi-relation" and might be what you need, but I find it suspicious that the multiplicities in $A$ and $B$ are not taken into account at all.