My understanding is that a multiset (roughly, a set where we care about multiplicity) can be modeled as a function $V\to\Bbb{Card}$ with set-sized support, where $V$ is the class of all sets, $\Bbb{Card}$ is the class of all cardinalities, and set-sized support means the class of sets on which the function is nonzero is a set. (These in turn can be modeled within ZFC as sets of input-output pairs where you simply ignore all the inputs whose outputs are zero.)
A similar and natural object would be a function $V\to\Bbb Z$ with set-sized support. (Again, $V$ is the class of all sets.) This would be like a multiset except that we can no longer have infinitely many copies of an element, but we may have negatively many copies of it.
These form a group and an algebra, which makes them particularly natural to work with. Adjoining the constant function $1$ to our algebra makes things especially nice (though it raises some issues in representing these within ZFC), since it allows us to say things like, "if the multiplicities of the elements of $X$ and $Y$ are all $1$ (i.e. they're pure sets), then $X\cap Y=XY$ and $X\cup Y=1-(1-X)(1-Y)$". The principle of inclusion-exclusion is essentially the equation\begin{align}X\cup Y\cup Z&=1-(1-X)(1-Y)(1-Z)\\&=X+Y+Z-XY-XZ-YZ+XYZ.\end{align}
In any case, my question is: do these natural objects (functions $V\to\Bbb Z$ with set-sized support, interpreted as variations on the multiset concept) have a name? If so, what is it?
To be clear, the sort of answer I'm expecting is a piece of terminology. I want a word that can fill the blank in the following piece of dialogue: "The right way to think about this puzzle is to frame it in terms of _____s instead of just sets."
(If there is no name, as I'm starting to worry is the case, I'd appreciate your help in inventing one - though I'd appreciate if you didn't just open a dictionary and choose a word at random.)