# 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?

• That's not a multiset, that's $\mathbb{Z}^S$, where $S$ represents the index set. – vadim123 May 5 '13 at 18:40
• @vadim123 Sorry, could you explain your comment a bit? Why is it not a multiset? – J.P. May 5 '13 at 19:05
• JP, because a multiset is $\mathbb{N}^S$. See @MJD's answer for more details. – vadim123 May 5 '13 at 20:37
• I imagine that it could be useful to represent members of $\mathbb Q$ as a multiset with possibly negative multiplicity indicating the prime factorization of both the numerator and denominator, where a negative multiplicity indicates that the factor is in the denominator. – dfan May 5 '13 at 23:39
• No, you're not allowed. – Jacob Wakem Dec 1 '16 at 4:54

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.
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.