nLab
multiset

Multisets

Idea

A multiset is like a set, just allowing that the elements have multiplicities. Thus the multiset {1,1,2}\{1,1,2\} differs from the multiset {1,2}\{1,2\}, while {1,2,1}\{1,2,1\} is the same as {1,1,2}\{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 𝒳=X,μ X\mathcal{X} = \langle X,\mu_X\rangle,can be defined as a set XX (its underlying set) together with a function μ X\mu_X (giving each element its multiplicity) from XX 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 XX 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 XX above).

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

A multisubset of UU is locally finite if every fibre is finite; it is finite if additionally the image of ff (which corresponds to AA in the original definition) is finite. A locally finite multisubset can also be described as a function from UU to the set of natural numbers; this is just the multiplicity function μ\mu again, now with UU (rather than XX) specified as the domain and allowing the value 00 to be taken.

Operations on multisets

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

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

  • The cardinality of a multiset is given by

    |𝒳|= eXμ X(e).|\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.

    𝒳𝒴=XY,min(μ X,μ Y).\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.

    𝒳𝒴=XY,max(μ X,μ Y).\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

    𝒳\𝒴={aXY|mu X(a)>μ Y(a)},μ Xμ Y. \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.

    𝒳+𝒴=XY,μ X+μ Y.\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

    𝒳𝒴=XY,μ Xμ Y.\mathcal{X}\mathcal{Y} = \langle X\cap Y, \mu_X\mu_Y\rangle.

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

  • The inner product of multisets is given by

    𝒳,𝒴= eXYμ X(e)μ Y(e).\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

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

Examples

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:AMSetF: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 UU considered as objects of Set/USet/U, a function from BUB\to U to BUB'\to U would be an arbitrary function BBB\to B' (not necessarily commuting with the projections to UU). But this doesn’t work if a multisubset of UU is an isomorphism class in Set/USet/U rather than merely an object of it. – Mike Shulman

Eric: This definition is taken from Syropoulos:

Definition

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

  1. The objects of the category consist of pairs (A,P)(A, P), where AA is a set and P:ASetP:A\to Set a presheaf on AA.
  2. If (A,P)(A,P) and (B,Q)(B,Q) are two objects of the category, an arrow between these objects is a pair (f,λ)(f,\lambda), where f:ABf:A\to B is a function and λ:PQf\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)(f,λ)(B,Q)(A,P)\stackrel{(f,\lambda)}{\to}(B,Q) and (B,Q)(g,μ)(C,R)(B,Q) \stackrel{(g,\mu)}{\to}(C,R) are arrows of the category, then (f,λ)(g,μ)=(gf,μ×λ)(f,\lambda)\circ (g,\mu) = (g\circ f, \mu\times\lambda), where gfg\circ f is the usual function composition and μ×λ:PR(gf)\mu\times\lambda:P\to R\circ (g\circ f).
  4. Given an object (A,P)(A,P), the identity arrow is (id A,id P)(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 UU are exactly objects of Set/USet/U. I’m surprised, though, that with his definition the only functions {1,1}{2,3}\{1,1\} \to \{2,3\} are constant; why can’t I send the two copies of 11 to different places?

Toby: If you could, then {1,1}\{1,1\} and {2,3}\{2,3\} would be isomorphic in this category, and we'd just have SetSet 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 MSetMSet 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 2Set^{\mathbf{2}}. Is that better?

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

Toby: At least Set 2Set^{\mathbf{2}} is different from SetSet. 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 2Set^{\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 𝒳\mathcal{X} is a pair (X,ρ)(X,\rho), where XX is a set and ρ\rho an equivalence relation on XX. The set XX is called the field of the real multiset. Elements of XX 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 𝒳=(X,ρ)\mathcal{X} = (X,\rho) and 𝒴=(Y,σ)\mathcal{Y} = (Y,\sigma), a morphism of real multisets is a function f:XYf:X\to Y which respects sorts; that is, if x,xXx,x'\in X and xρxx \rho x', then f(x)σf(x)f(x)\sigma f(x').

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

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

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

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

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

As a writer (perhaps Toby) was saying above, a locally finite multiset MM can be thought of as an ordinary set XX equipped with a multiplicity function μ:X\mu: X \to \mathbb{N}. A multisubset of MM can then be reckoned as XX equipped with a function ν:X\nu: X \to \mathbb{N} which is bounded above by μ\mu. To take the intersection of two multisubsets ν,ν:X\nu, \nu': X \to \mathbb{N}, you take the minimum or inf of ν,ν\nu, \nu'. Your question can then be translated to one where X={1,2,3}X = \{1, 2, 3\}, where ν(1)=2,ν(2)=1,ν(3)=0\nu(1) = 2, \nu(2) = 1, \nu(3) = 0 and ν(1)=2,ν(2)=0,ν(3)=1\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 UU? It seems like a multiset X,μ\langle X,\mu\rangle (XX s a set and μ:X\mu:X\to\mathbb{N}) can be thought of as a vector with μ\mu providing the coefficients.

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

𝒳=3{1}+1{2}+4{3}.\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 μ:X\mu\colon X \to \mathbb{N}. If you allow arbitrary cardinalities, then it's the free module on UU over the rig of cardinal numbers.

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