Multisets

Idea

A multiset is like a set, just allowing that the elements have multiplicities. Thus the multiset $\{1,1,2\}$ differs from the multiset $\{1,2\}$, while $\{1,2,1\}$ is the same as $\{1,1,2\}$. Multisets are useful in combinatorics. See wikipedia.

Definitions

While it is possible to take multisets as a fundamental concept in foundations, it is more common to define them in terms of sets and functions.

A multiset $\mathcal{X} = \langle X,\mu_X\rangle$,can be defined as a set $X$ (its underlying set) together with a function $\mu_X$ (giving each element its multiplicity) from $X$ to a class of nonzero cardinal numbers. A multiset is locally finite if multiplicity takes values in the natural numbers. Many authors take all multisets to be locally finite; that is the default in combinatorics. The multiset is finite if it is locally finite and $X$ is a finite set. We can also define a multiset to be a function from the proper class of all objects to the class of all cardinal numbers, with the proviso that the objects whose multiplicity is nonzero form a set (the set $X$ above).

If we are only interested in multisets with elements drawn from a given set $U$ (as is common in combinatorics), then an alternative definition is very useful: a multisubset of $U$ is a function $f\colon B \to U$, where two multisubsets $f\colon B \to U$ and $f'\colon B' \to U$ are considered equal if there is a bijection $g\colon B \to B'$ that makes a commutative triangle. In other words, a multisubset of $U$ is an isomorphism class in the slice category $Set/U$. (Compare this to the structural definition of subset of $U$ as an injective function to $U$.)

A multisubset of $U$ is locally finite if every fibre is finite; it is finite if additionally the image of $f$ (which corresponds to $A$ in the original definition) is finite. A locally finite multisubset can also be described as a function from $U$ to the set of natural numbers; this is just the multiplicity function $\mu$ again, now with $U$ (rather than $X$) specified as the domain and allowing the value $0$ to be taken.

Operations on multisets

The operations on cardinal numbers induce operations on multisets (or on multisubsets of any given set $U$).

In the following, let $\mathcal{X} = \langle X,\mu_X\rangle$ and $\mathcal{Y} = \langle Y,\mu_Y\rangle$ be multisets.

• The cardinality of a multiset is given by

$|\mathcal{X}| = \sum_{e\in X} \mu_X(e).$
• The intersection of multisets is the multiset whose cardinality is given by the infimum operation on cardinal numbers.

$\mathcal{X}\cap\mathcal{Y} = \langle X\cap Y, \min(\mu_X,\mu_Y)\rangle.$
• The union of multisets is the multiset whose cardinality is given by the supremum operation on cardinal numbers.

$\mathcal{X}\cup\mathcal{Y} = \langle X\cup Y, \max(\mu_X,\mu_Y)\rangle.$
• The set difference of multisets is the multiset given by

$\mathcal{X} \backslash \mathcal{Y} = \langle \{ a \in X \cup Y \;|\; mu_X(a) \gt \mu_Y(a) \}, \mu_X - \mu_Y \rangle .$
• The sum of multisets is the multiset whose cardinality is given by addition of cardinal numbers; this has no analogue for ordinary sets.

$\mathcal{X}+\mathcal{Y} = \langle X\cup Y, \mu_X+\mu_Y\rangle.$
• The product of multisets (turning them into a rig) is the multiset whose cardinality is given by the product of cardinal numbers

$\mathcal{X}\mathcal{Y} = \langle X\cap Y, \mu_X\mu_Y\rangle.$

Note that if $\mathcal{X}$ is a set, then $\mathcal{X}\mathcal{X} = \mathcal{X}.$

• The inner product of multisets is given by

$\langle\mathcal{X},\mathcal{Y}\rangle = \sum_{e\in{X\cap Y}} \mu_X(e) \mu_Y(e).$

Note that the inner product corresponds to the cardinality of the product

$\langle\mathcal{X},\mathcal{Y}\rangle = |\mathcal{X}\mathcal{Y}|.$

References

Is there a reason that you moved these references up here? We need them especially for the stuff about morphisms below. —Toby

Discussion

Eric: What would a colimit over an MSet-valued functor $F:A\to MSet$ look like?

Toby: That depends on what the morphisms are.

Eric: I wonder if there is enough freedom in the definition of morphisms of multisets so that the colimit turns out particularly nice. I’m hoping that it might turn out to be simply the sum of multisets. According to limits and colimits by example the colimit of a Set-valued functor is a quotient of the disjoint union.

Toby: I think that you might hope for the coproduct (but not a general colimit) of multisets to be a sum rather than a disjoint union. Actually, you could argue that the sum is the proper notion of disjoint union for abstract multisets.

Morphisms

What is a function between multisets? I would be inclined to say that for multisubsets of an ambient universe $U$ considered as objects of $Set/U$, a function from $B\to U$ to $B'\to U$ would be an arbitrary function $B\to B'$ (not necessarily commuting with the projections to $U$). But this doesn’t work if a multisubset of $U$ is an isomorphism class in $Set/U$ rather than merely an object of it. – Mike Shulman

Eric: This definition is taken from Syropoulos:

Definition

Category $\mathbf{MSet}$ is a category of all possible multisets.

1. The objects of the category consist of pairs $(A, P)$, where $A$ is a set and $P:A\to Set$ a presheaf on $A$.
2. If $(A,P)$ and $(B,Q)$ are two objects of the category, an arrow between these objects is a pair $(f,\lambda)$, where $f:A\to B$ is a function and $\lambda:P\to Q\circ f$ is a natural transformation, i.e., a family of functions.
3. Arrows compose as follows: suppose that $(A,P)\stackrel{(f,\lambda)}{\to}(B,Q)$ and $(B,Q) \stackrel{(g,\mu)}{\to}(C,R)$ are arrows of the category, then $(f,\lambda)\circ (g,\mu) = (g\circ f, \mu\times\lambda)$, where $g\circ f$ is the usual function composition and $\mu\times\lambda:P\to R\circ (g\circ f)$.
4. Given an object $(A,P)$, the identity arrow is $(id_A, id_P)$.

The last part of the definition is a kind of wreath product (see [4]). However, it is not clear at the moment how this definition fits into the general theory of wreath products.

Mike Shulman: Huh. So his definition takes a multiset to assign a set to every element, rather than a cardinality to every element, so that the multisubsets of $U$ are exactly objects of $Set/U$. I’m surprised, though, that with his definition the only functions $\{1,1\} \to \{2,3\}$ are constant; why can’t I send the two copies of $1$ to different places?

Toby: If you could, then $\{1,1\}$ and $\{2,3\}$ would be isomorphic in this category, and we'd just have $Set$ back again.

I find Mathematics of Multisets especially interesting for its distinction between multisets with distinguishable objects and ‘pure’ multisets with indistinguishable objects. The definition above involving cardinal numbers gives us ‘pure’ multisets, unlike the objects of the category $MSet$ above.

Normally, one only needs multisubsets of a given set, and one is not interested in functions between them. But if one wants to make a category of abstract multisets, then the pure and impure versions are different!

Mike Shulman: Whereas his definition makes the category of multisets equivalent to $Set^{\mathbf{2}}$. Is that better?

I don’t find it wrong that the category of multisets would be equivalent to $Set$, since $Set$ only sees “structural” properties of sets, and the fact that two elements of a set are “the same” (which is what distinguishes $\{1,1\}$ from $\{2,3\}$) is a nonstructural property that only makes sense in the context of sub-multisets of some ambient set.

Toby: At least $Set^{\mathbf{2}}$ is different from $Set$. And how do you decide whether being ‘the same’ is a structural property of a multiset? We're trying to take an idea that originally applied only to collections of elements from a fixed universe and move it to a more abstract settings; there are (at least) two ways to do that, and Syropoulos has chosen the more interesting one. (Anyway, if somebody asked me to come up with a structural notion of abstract multiset, the first thing that I would think of —and did think of, before this discussion started— is an object of $Set^{\mathbf{2}}$.) Asking which notion is correct is not really a fair question.

Eric: The paper Mathematics of Multisets is worth having a look. I might have pasted a suboptimal piece. He talks about two types of multisets (and more actually): 1.) real multisets and 2. multisets. Here is another quote:

Real multisets and multisets are associated with a (ordinary) set and an equivalence relation or a function, respectively. Here are the formal definitions:

Definition 1. A real multiset $\mathcal{X}$ is a pair $(X,\rho)$, where $X$ is a set and $\rho$ an equivalence relation on $X$. The set $X$ is called the field of the real multiset. Elements of $X$ in the same equivalence class will be said to be of the same sort; elements in different equivalence classes will be said to be of different sorts.

Given two real multisets $\mathcal{X} = (X,\rho)$ and $\mathcal{Y} = (Y,\sigma)$, a morphism of real multisets is a function $f:X\to Y$ which respects sorts; that is, if $x,x'\in X$ and $x \rho x'$, then $f(x)\sigma f(x')$.

Definition 2. Let $D$ be a set. A multiset over $D$ is just a pair $\langle D, f\rangle$, where $D$ is a set and $f:D\to\mathbb{N}$ is a function.

The previous definition is the characteristic function definition method for multisets.

Remark 1. Any ordinary set $A$ is actually a multiset $\langle A,\chi_A\rangle$, where $\chi_A$ is its characteristic function.

Eric: Given $X = \{1,1,2\}$ and $Y = \{1,1,3\}$, is $X\cap Y = \{1,1\}$ or is $X\cap Y = \{1\}$?

Todd: It’s $\{1, 1\}$. (To make the question structural, we should think of $X$ and $Y$ as multisubsets of some other multiset, but never mind.)

As a writer (perhaps Toby) was saying above, a locally finite multiset $M$ can be thought of as an ordinary set $X$ equipped with a multiplicity function $\mu: X \to \mathbb{N}$. A multisubset of $M$ can then be reckoned as $X$ equipped with a function $\nu: X \to \mathbb{N}$ which is bounded above by $\mu$. To take the intersection of two multisubsets $\nu, \nu': X \to \mathbb{N}$, you take the minimum or inf of $\nu, \nu'$. Your question can then be translated to one where $X = \{1, 2, 3\}$, where $\nu(1) = 2, \nu(2) = 1, \nu(3) = 0$ and $\nu'(1) = 2, \nu'(2) = 0, \nu'(3) = 1$.

Eric: Thanks Todd! The reference Mathematics of Multisets explains this nicely too.

Eric: What is the difference (aside from negatives) between multisets and abelian groups freely generate by some set $U$? It seems like a multiset $\langle X,\mu\rangle$ ($X$ s a set and $\mu:X\to\mathbb{N}$) can be thought of as a vector with $\mu$ providing the coefficients.

For example, we could express the multiset $\mathcal{X} = \{1,1,1,2,3,3,3,3,3\}$ as

$\mathcal{X} = 3\{1\} + 1\{2\} + 4\{3\}.$

Toby: The only difference is notation; see the note at the end of inner product of multisets.

Mike Shulman: At least, if all your multisets are locally finite.

Toby: Right; which they are for Eric, who specified $\mu\colon X \to \mathbb{N}$. If you allow arbitrary cardinalities, then it's the free module on $U$ over the rig of cardinal numbers.

Revised on May 31, 2012 19:19:40 by Toby Bartels (64.89.53.71)