Is there such a thing as a multiset with a "negative" number of some element? Is it possible for a multiset to have a "negative" number of one or more elements?  If so, how are such multisets defined, and what terminology exists for them?
 A: As I pointed out a while back, there isn't much standard terminology or notation even for ordinary multisets. Mathematicians usually resort to one of two workarounds when they need to deal with multisets: they either replace the multiset of elements of $S$ with a (not strictly) monotonic sequence of elements of $S$, or they replace it with a function $c:S\to\Bbb N$ that counts how many of each element of $S$ there is in the multiset. For example, $c(\bullet) = 3$ means that the multiset $c$ contains three instances of the element $\bullet$.
As vadim123 pointed out in a comment, it's easy to adjust the latter workaround from $c:S\to\Bbb N$ to $c:S\to\Bbb Z$, but at that point the object you are dealing with is a lot more like a function than it is like a set, and it's not clear what benefit you would get from trying to think of it like some weird kind of set.
A: Yes, such multisets with negative multiplicities are considered and used. For example, see this link .
A: One way to describe such multisets, at least those which only consist of a finite number of elements, is as elements of the free abelian group on a set $S$ (this is not the same as the set of functions $S \to \mathbb{Z}$ when $S$ is infinite, and in any case it is a covariant functor rather than a contravariant one). This is a common construction in algebraic topology and algebraic geometry. 
