## Multiset operations as binary operations (part 1)

June 9, 2008

Review: Associativity, commutativity, and idempotence

A binary operation $\cdot$ on a set $X$ is a function $\cdot: X\times X \to X$. It is usually written using an infix notation $x\cdot y$ rather than a functional notation such as $\cdot\left(x, y\right)$.

The operation $\cdot$ is associative if and only if $a\cdot\left(b\cdot c\right) = \left(a\cdot b\right)\cdot c$ for all $a,b,c\in X$. It is commutative if and only if $a\cdot b = b\cdot a$ for all $a,b\in X$. It is idempotent if and only if $a\cdot a = a$ for all $a\in X$.

We’ve defined four operations on families of multisets $\left\{\mathcal{M}_i\right\}_{i\in I}$ where $\mathcal{M}_i \in \mathbf{MSet}_X$ for all $i\in I$: sums, products, intersections, and unions. As I’ve already commented in a couple of places, we can restrict our attention to families where $\left|I\right| = 2$. This provides us with four binary operations on $\mathbf{MSet}_X$. Today’s post collects together the properties of these binary operations.

Proposition 1: The operations $\uplus$, $\:\cdot\!\!\!\!\cup$, $\cap$, and $\cup$ on $\mathbf{MSet}_X$ are all associative and commutative.

Proof:

Let $\mathcal{M}_1, \mathcal{M}_2, \mathcal{M}_3 \in \mathbf{MSet}_X$. The first four parts follow directly from the associativity and commutativity of $+$ and $\cdot$ as binary operations on $\mathbf{Card}$.

Associativity of $\uplus$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \uplus \left(\mathcal{M}_2 \uplus \mathcal{M}_3\right) & \left(X, f_1 + \left(f_2 + f_3\right)\right) \\ & \left(X, \left(f_1 + f_2\right) + f_3\right) \\ & \left(\mathcal{M}_1 \uplus \mathcal{M}_2\right) \uplus \mathcal{M}_3\end{array}$

Commutativity of $\uplus$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \uplus \mathcal{M}_2& \left(X, f_1 + f_2\right) \\ & \left(X, f_2 + f_1\right) \\ & \mathcal{M}_2 \uplus \mathcal{M}_1\end{array}$

Associativity of $\:\cdot\!\!\!\!\cup$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \:\cdot\!\!\!\!\cup\: \left(\mathcal{M}_2 \:\cdot\!\!\!\!\cup\: \mathcal{M}_3\right) & \left(X, f_1 \cdot \left(f_2 \cdot f_3\right)\right) \\ & \left(X, \left(f_1 \cdot f_2\right) \cdot f_3\right) \\ & \left(\mathcal{M}_1 \:\cdot\!\!\!\!\cup\: \mathcal{M}_2\right) \:\cdot\!\!\!\!\cup\: \mathcal{M}_3\end{array}$

Commutativity of $\:\cdot\!\!\!\!\cup$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \:\cdot\!\!\!\!\cup\: \mathcal{M}_2& \left(X, f_1 \cdot f_2\right) \\ & \left(X, f_2 \cdot f_1\right) \\ & \mathcal{M}_2 \:\cdot\!\!\!\!\cup\: \mathcal{M}_1\end{array}$

The remaining parts follow from the following facts:

1. Let $A, B$ be partially ordered sets. Then $\inf \left(A \cup \left\{\inf B\right\}\right) = \inf\left(A\cup B\right)$ and $\sup \left(A \cup \left\{\sup B\right\}\right) = \sup \left(A\cup B\right)$ whenever the suprema and infima exist.
2. The supremum and infimum of every set of cardinals exist.

Associativity of $\cap$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \cap \left(\mathcal{M}_2 \cap \mathcal{M}_3\right) & \left(X, \inf \left\{f_1, \inf\left\{f_2, f_3\right\}\right\}\right) \\ & \left(X, \inf\left\{f_1, f_2, f_3\right\}\right) \\ & \left(X, \inf \left\{\inf\left\{f_1, f_2\right\}, f_3\right\}\right) \\ & \left(\mathcal{M}_1 \cap\mathcal{M}_2\right) \cap \mathcal{M}_3\end{array}$

Commutativity of $\cap$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \cap \mathcal{M}_2& \left(X, \inf \left\{f_1, f_2\right\}\right) \\ & \mathcal{M}_2 \cap \mathcal{M}_1\end{array}$

Associativity of $\cup$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \cup \left(\mathcal{M}_2 \cup \mathcal{M}_3\right) & \left(X, \sup \left\{f_1, \sup\left\{f_2, f_3\right\}\right\}\right) \\ & \left(X, \sup\left\{f_1, f_2, f_3\right\}\right) \\ & \left(X, \sup \left\{\sup\left\{f_1, f_2\right\}, f_3\right\}\right) \\ & \left(\mathcal{M}_1 \cup\mathcal{M}_2\right) \cup \mathcal{M}_3\end{array}$

Commutativity of $\cup$.

$\begin{array}{r@{\:=\:}l} \mathcal{M}_1 \cup \mathcal{M}_2& \left(X, \sup \left\{f_1, f_2\right\}\right) \\ & \mathcal{M}_2 \cup \mathcal{M}_1\end{array}$

Proposition 2: The operations $\cap$ and $\cup$ on $\mathbf{MSet}_X$ are idempotent.

Proof:

Let $\mathcal{M}\in \mathbf{MSet}_X$.

Idempotence of $\cap$.

$\begin{array}{r@{\:=\:}l} \mathcal{M} \cap \mathcal{M}& \left(X, \inf \left\{f, f\right\}\right) \\ & \left(X, f\right) \\ & \mathcal{M}\end{array}$

Idempotence of $\cup$.

$\begin{array}{r@{\:=\:}l} \mathcal{M} \cup \mathcal{M}& \left(X, \sup \left\{f, f\right\}\right) \\ & \left(X, f\right) \\ & \mathcal{M}\end{array}$

The other two operations—sum and product—are not idempotent. One counterexample is the multiset $\mathcal{M} = \left\{a^3\right\}$, for which we have

$\mathcal{M}\uplus\mathcal{M} = \left\{a^6\right\} \neq \mathcal{M}$

and

$\mathcal{M}\:\cdot\!\!\!\!\cup\:\mathcal{M} = \left\{a^9\right\} \neq \mathcal{M}$.

## Properties of multiset union

June 4, 2008

We defined the union of a family of multisets last week. Today we’re going to look at some basic properties of the union operation on $\mathbf{MSet}_X$.

Suppose $\left\{\mathcal{M}_i = \left(X, f_i\right)\right\}_{i\in I}$ is a family of multisets over $X$. Then the following relationships hold:

(i) $\mathrm{support}\left(\bigcup_{i\in I}{\mathcal{M}_i}\right) = \bigcup_{i\in I}{\mathrm{support}\left(\mathcal{M}_i\right)}$.

(ii) $\mathcal{M}_i \subseteq \bigcup_{i\in I}{\mathcal{M}_i}$ for all $i\in I$.

(iii) $\bigcup_{i\in I}{\mathcal{M}_i} \subseteq \biguplus_{i\in I}{\mathcal{M}_i}$.

(iv) $\bigcup_{i\in I}{\mathcal{M}_i} \subseteq \:\cdot\!\!\!\!\!\!\;\bigcup_{i\in I}{\mathcal{M}_i}$ provided that all of the multisets $\mathcal{M}_i$ have the same support.

(v) $\bigcap_{i\in I}{\mathcal{M}_i} \subseteq \bigcup_{i\in I}{\mathcal{M}_i}$.

The proof of part (i), like the similar result for multiset intersections, is a straightforward series of equivalencies:

$\begin{array}{r@{\:\Leftrightarrow\:}l} x\in \mathrm{support}\left(\bigcup_{i\in I}{\mathcal{M}_i}\right) & \sup f_i(x) \neq 0 \\ & \left(\exists i\in I\right)\: f_i\left(x\right)\neq 0 \\ & \left(\exists i\in I\right)\: x\in\mathrm{support}\left(\mathcal{M}_i\right) \\ & x\in\bigcup_{i\in I}{\mathrm{support}\left(\mathcal{M}_i\right)}. \end{array}$

Part (ii) follows directly from the definition of the supremum of a set, since $f_i\left(x\right) \leq \sup f_i\left(x\right)$ for all $x\in X$ and $i\in I$.

To prove part (iii), start by recalling that the sum of a family of cardinals $\left\{\kappa_i\right\}_{i\in I}$ is given by

$\sum_{i\in I}{\kappa_i} = \left|\bigcup_{i\in I}{A_i}\right|$

where the sets $A_i$ are disjoint and $\kappa_i = \left|A_i\right|$ for all $i\in I$. Therefore, letting $A_{i,x}$ be a family of disjoint sets such that $\left|A_{i,x}\right| = f_i\left(x\right)$, we have

$\kappa_x = \sum_{i\in I}{f_i\left(x\right)} = \left|\bigcup_{i\in I}{A_{i,x}}\right| \geq \left|A_{i,x}\right| = f_i\left(x\right)$

for all $i\in I$ and $x\in X$. This establishes that $\kappa_x$ is an upper bound for $f_i\left(x\right)$ in the linearly ordered set $\left(\mathbf{Card}, \leq\right)$, and so $\kappa_x \geq \sup f_i\left(x\right)$. The result in part (iii) follows immediately since this holds for all $x\in X$.

The proof of part (iv) is similar, but relies on the definition of the product of a family of cardinals. The product of a family of cardinals $\left\{\kappa_i\right\}_{i\in I}$ is given by

$\prod_{i\in I}{\kappa_i} = \left|\prod_{i\in I}{A_i}\right|$

where $\kappa_i = \left|A_i\right|$ for all $i\in I$. Letting $A_{i,x}$ be a family of sets such that $\left|A_{i,x}\right| = f_i\left(x\right)$, we have

$\kappa_x = \prod_{i\in I}{f_i\left(x\right)} = \left|\prod_{i\in I}{A_{i,x}}\right| \geq \left|A_{i,x}\right| = f_i\left(x\right)$

for all $i\in I$ and $x\in X$, provided that $A_{i,x} \neq \emptyset$ for all $i\in I$. This establishes that $\kappa_x$ is an upper bound for $f_i\left(x\right)$ in the linearly ordered set $\left(\mathbf{Card}, \leq\right)$ whenever $x\in\mathrm{support}\left(\bigcup_{i\in I}{\mathcal{M}_i}\right)$, which is the common support of all multisets in the family $\left\{\mathrm{M}_i\right\}_{i\in I}$. If $x$ is not in this support, then $\prod_{i\in I}{f_i\left(x\right)} = 0 = \sup f_i\left(x\right)$. In either case, $\kappa_x \geq \sup f_i\left(x\right)$, and part (iv) follows immediately since this holds for all $x\in X$.

The requirement that all multisets in the family have the same support is necessary; you can see this by considering the product and union of the multisets $\left\{a\right\}$ and $\left\{b\right\}$.

Finally, part (v) follows immediately from the fact that

$\inf f_i\left(x\right) \leq f_i\left(x\right) \leq \sup f_i\left(x\right)$

for all $i\in I$ and $x\in X$.

## Multiset union

May 28, 2008

Over the past two weeks, we’ve introduced three operations on families of multisets: sums, products, and intersections. The other basic operation on families of multisets, unions, is today’s topic. As we did with the other operations, we’ll revisit the characteristic function and then adapt this to the context of multisets.

The union of two sets $A, B\in \wp\left(X\right)$ can, as we’ve already seen, be represented in terms of the characteristic function as

$C=A\cup B \:\Leftrightarrow\: \chi_C = \max\left(\chi_A, \chi_B\right)$

where the maximum is taken in the ordered field $\mathbb{F}_2$ (just as we took the minimum in this field when looking at the intersection). This maximum is, in fact, a supremum:

$\sup\left\{\chi_A\left(x\right), \chi_B\left(x\right)\right\}$ for each $x\in X$.

For any family of functions $f_i: X\to Y$, we let

$\begin{array}{lrl}\sup \left\{f_i\right\} := &g:& X\to Y \\ &&x\mapsto \sup \left\{f_i\left(x\right)\right\} \end{array}$

provided that the supremum exists for each $x\in X$. Since $\mathbb{F}_2$ is finite—which ensures the existence of the suprema—and the characteristic functions are taken over a common domain, the representation of union in terms of characteristic functions extends to any family of subsets of $X$. That is, for any family $\left\{A_i\right\}_{i\in I}$ where each $A_i \in \wp\left(X\right)$, we have

$A = \bigcup_{i\in I}{A_i} \:\Leftrightarrow\: \chi_A = \sup\left\{\chi_{A_i}\right\}$.

This is completely analogous to the situation we encountered with the intersection of a family of subsets of $X$, including the avoidance of the algebraic properties of $\mathbb{F}_2$.

With the intersection, we were able to construct a definition on $\mathbf{MSet}_X$ that was analogous to the definition on $\mathbf{SSet}_X$ because both codomains—$\mathbb{F}_2$ and $\mathbf{Card}$—were well-ordered, so the necessary infima were guaranteed to exist. In $\mathbb{F}_2$, we know that the suprema exist because the set is linearly ordered and finite. $\mathbf{Card}$ is linearly ordered, but not finite. This raises a question: does every set of cardinals have a supremum?

The answer to this question depends on the set theory in which one is working. In our case, the axiom of choice allows us to actually define cardinals in terms of ordinals; in particular, cardinals are defined to be the initial ordinals. An ordinal is an initial ordinal if it is not equinumerous with any smaller ordinal. Moreover, every set of ordinals has a supremum. Since cardinals are ordinals, any set $\mathcal{K}$ of cardinals has an ordinal supremum $\omega$. The cardinal $\kappa$ which is equinumerous with $\omega$ is the cardinal supremum of $\mathcal{K}$.

With this in hand, we’re ready to define the union of a family of multisets.

Definition

Let $\mathrm{M} = \left\{\mathcal{M}_i = \left(X, f_i\right)\right\}_{i\in I}$ be a family of multisets over a set $X$. The multiset union of $\mathrm{M}$ is the set

$\mathcal{M} = \bigcup_{i\in I}{\mathcal{M}_i} := \left(X, \sup\left\{f_i\right\}\right)$.

The multiplicity functions $f_i$ are defined on a common domain, and every set of cardinals has a supremum. This ensures that the multiset union is well-defined on $\mathbf{MSet}_X$. We’ll deal with the properties of this operation in my next post on multisets.

## Properties of multiset intersection

May 27, 2008

Today, we’ll examine the properties of the multiset intersection that we defined yesterday.

Suppose $\left\{\mathcal{M}_i = \left(X, f_i\right)\right\}_{i\in I}$ is a family of multisets over $X$. Then the following relationships hold:

(i) $\mathrm{support}\left(\bigcap_{i\in I}{\mathcal{M}_i}\right) = \bigcap_{i\in I}{\mathrm{support}\left(\mathcal{M}_i\right)}$.

(ii) $\bigcap_{i\in I}{\mathcal{M}_i} \subseteq \mathcal{M}_i$ for all $i\in I$.

(iii) $\bigcap_{i\in I}{\mathcal{M}_i} \subseteq \biguplus_{i\in I}{\mathcal{M}_i}$.

(iv) $\bigcap_{i\in I}{\mathcal{M}_i} \subseteq \:\cdot\!\!\!\!\!\!\;\bigcup_{i\in I}{\mathcal{M}_i}$.

The proof of part (i) is a straightforward series of equivalencies:

$\begin{array}{r@{\:\Leftrightarrow\:}l} x\in \mathrm{support}\left(\bigcap_{i\in I}{\mathcal{M}_i}\right) & \inf f_i(x) \neq 0 \\ & \left(\forall i\in I\right)\: f_i\left(x\right)\neq 0 \\ & \left(\forall i\in I\right)\: x\in\mathrm{support}\left(\mathcal{M}_i\right) \\ & x\in\bigcap_{i\in I}{\mathrm{support}\left(\mathcal{M}_i\right)}. \end{array}$

Part (ii) follows directly from the definition of the infimum of a set, since $\inf f_i\left(x\right) \leq f_i\left(x\right)$ for all $x\in X$ and $i\in I$.

Recalling that the cardinal sum is a monotonic nondecreasing operation, we see that

$\inf f_i\left(x\right) \leq f_i\left(x\right) \leq \sum_{i\in I}{f_i\left(x\right)}$

for all $x\in X$, which proves part (iii).

Part (iv) requires only slightly more work. To begin with, consider a family $\left\{\kappa_i\right\}_{i\in I}$ of cardinals. If there exists some $i\in I$ such that $\kappa_i = 0$, then $\prod_{i\in I}{\kappa_i} = 0$. If, however, no such $i\in I$ exists, then

$\kappa_i\leq\prod_{i\in I}{\kappa_i}$ for all $i\in I$.

(This relies on the axiom of choice, which we have been assuming from the outset.) In other words, the product of any family of nonzero cardinals is monotonic nondecreasing.

Let $x\in\bigcap_{i\in I}{\mathcal{M}_i}$. If there exists some $i\in I$ such that $x\not\in\mathcal{M}_i$, then the multiplicity of $x$ in the intersection would be 0, contradicting the fact that $x$ is a member of the intersection. Since $f_i\left(x\right)\neq 0$ for all $i\in I$, we have

$\inf f_i\left(x\right) \leq f_i\left(x\right) \leq \prod_{i\in I}{f_i\left(x\right)}$.

For $x\not\in\bigcap_{i\in I}{\mathcal{M}_i}$, we have

$\inf f_i\left(x\right) = 0 = \prod_{i\in I}{f_i\left(x\right)}$.

In either case, $\inf f_i\left(x\right) \leq \prod_{i\in I}{f_i\left(x\right)}$ for all $x\in X$, and (iv) holds.

## Multiset intersection

May 26, 2008

Last Monday, I mentioned that multiset products are not the best available extension of the intersection of a family of sets so that it applies to multisets. Now that we’ve talked about the concepts of infima and well-ordering, we’re ready to define multiset intersections.

A bit further back, we discussed how characteristic functions can be used to represent subsets of a set $X$ and the operations on $\wp\left(X\right)$. At the time, we noted that the intersection of two sets $A, B\in \wp\left(X\right)$ is represented in terms of the characteristic function as

$C=A\cap B \:\Leftrightarrow\: \chi_C = \min\left(\chi_A,\chi_B\right)$

where the minimum is taken in the ordered field $\mathbb{F}_2$. For each $x\in X$, this minimum is just $\inf\left\{\chi_A\left(x\right), \chi_B\left(x\right)\right\}$.

Given a family of functions $f_i: X\to Y$, we let

$\begin{array}{lrl}\inf \left\{f_i\right\} := &g:& X\to Y \\ &&x\mapsto \inf \left\{f_i\left(x\right)\right\} \end{array}$.

Since $\mathbb{F}_2$ is well-ordered and the characteristic functions are taken over a common domain, our representation of intersection in terms of characteristic functions extends to any family of subsets of $X$. That is, for any family $\left\{A_i\right\}_{i\in I}$ where each $A_i \in \wp\left(X\right)$, we have

$A = \bigcap_{i\in I}{A_i} \:\Leftrightarrow\: \chi_A = \inf\left\{\chi_{A_i}\right\}$.

We also saw a representation of intersections as the product of the characteristic functions. When dealing with sets in $\wp\left(X\right)$, the two representations correspond to the same objects and operations in the class $\mathbf{SSet}_X$. However, there is an important difference between the two: the representation in terms of products relies (exclusively) on the algebraic properties of the codomain, while the representation in terms of infima relies (exclusively) on the order properties of the codomain.

The algebraic properties of the field $\mathbb{F}_2$ and arithmetic on $\mathbf{Card}$ are quite different. However, both of them are well-ordered classes, so their order properties are similar in many respects. This leads us to the following definition:

Definition

Let $\mathrm{M} = \left\{\mathcal{M}_i = \left(X, f_i\right)\right\}_{i\in I}$ be a family of multisets over a set $X$. The multiset intersection of $\mathrm{M}$ is the set

$\mathcal{M} = \bigcap_{i\in I}{\mathcal{M}_i} := \left(X, \inf\left\{f_i\right\}\right)$.

The multiplicity functions $f_i$ are defined on a common domain, and $\mathbf{Card}$ is well-ordered by the usual relation $\leq$. Just as we found with the characteristic function, these facts are sufficient to ensure that the multiset intersection is always well-defined on $\mathbf{MSet}_X$.

As usual, when $\mathrm{M}$ contains only two multisets, we will use the infix notation $\mathcal{M}_1 \cap \mathcal{M}_2$ for the intersection. In this case, $\cap$ is an associative and commutative binary operation on $\mathbf{Mset}_X$. Next time, we’ll look at some of the properties of multiset intersections.

## Multiset products

May 19, 2008

When we looked at the characteristic function for the subset of a set, we started by examining both sums and products of the characteristic function in $\mathbb{F}_2.$ Last week, we looked at the multiset sum, which we defined in terms of the cardinal sum that we’ve been using for a while now. To consider the multiset analogue of the products of characteristic functions, we need another bit of cardinal arithmetic.

Definition

Let $I$ be a set, $\left\{\kappa_i\right\}_{i\in I}$ be a family of cardinals, and $\left\{A_i\right\}_{i\in I}$ be a family of sets such that $\left|A_i\right| = \kappa_i$ for $i\in I$. The product of this family of cardinals is

$\prod_{i\in I} \kappa_i := \left| \prod_{i\in I} {A_i} \right|.$

For infinite index sets $i\in I,$ the axiom of choice ensures that the set product $\mathcal{A} = \prod_{i\in I}{A_i}$ is not empty. As with the sum, it is possible to prove that $\left|\mathcal{A}\right|$ does not depend on the choice of sets $A_i,$ and so the product is well-defined.

Definition

Let $\mathcal{M} = \left\{\mathcal{M}_i = \left(X, f_i\right)\right\}_{i\in I}$ be an indexed family of multisets. The multiset product of $\mathcal{M}$ is the multiset

$\cdot\!\!\!\!\!\!\:\bigcup_{i\in I} \mathcal{M}_i := \left(X, \prod_{i\in I}f_i\right).$

As we saw with the multiset sum, $\mathrm{dom}\left(f_i\right) = X$ for all $i\in I.$ The cardinal product on the right is therefore well-defined, and the result is again a multiset over $X.$ Unsurprisingly, we also use the infix notation $\mathcal{M}_1 \cdot\!\!\!\!\cup\, \mathcal{M}_2$ when dealing with just a pair of multisets. This binary operation on $\mathbf{MSet}_X$ is both commutative and associative.

The cardinal product satisfies the equation $0\cdot\kappa = 0$ for all $\kappa\in\mathbf{Card}.$ It follows that $x\in\mathrm{support}\left(\mathcal{M}\right)$ if and only if $f_i(x) \neq 0$ for all $i\in I.$ This gives us the following identity:

$\mathrm{support}\left(\;\cdot\!\!\!\!\!\!\;\bigcup_{i\in I} \mathcal{M}_i\right) = \bigcap_{i\in I} \mathrm{support}\left(\mathcal{M}_i\right).$

As a binary operation on $\mathbf{MSet}_X,$ the multiset product is not in general idempotent. Unlike the multiset sum, however, the class of multisets $\left(X, f_i\right)$ where $f_i\left(X\right)\subseteq\left\{0,1\right\}$ is closed under multiset products. For this class, it is even idempotent. Even more suggestive is the fact that $x\in \mathcal{M}_1 \cdot\!\!\!\!\cup\, \mathcal{M}_2$ if and only if $x\in\mathcal{M}_1$ and $x\in\mathcal{M}_2.$

These observations raise a question: is the multiset product a good candidate to be viewed as an extension to multisets of the intersection of sets? Let’s consider the set $X = \left\{a, b, c\right\}$ and the multisets $\mathcal{M}_1 = \left\{a^2, b^4, c^2\right\}$ and $\mathcal{M}_1 = \left\{a, b^0, c^5\right\}$ (the expression $x^\kappa$ indicates that $\kappa = f\left(x\right)$ is the multiplicity of $x$ in the given multiset). Here, we have

$\mathcal{M}_1 \cdot\!\!\!\!\cup\, \mathcal{M}_2 = \left\{a^2, b^0, c^{10}\right\} \not\subseteq \mathcal{M}_1.$

This is contrary to the intuitive idea that an intersection should always be contained in every multiset that participates in the intersection. Counterintuitive results aren’t always wrong. Given the notation, however, we can easily guess that we will soon see an operation that is a better choice to consider as an intersection of multisets.

## Multiset sums

May 14, 2008

On Monday, we considered how the sets $A\in\wp\left(X\right)$ can be represented as functions $\chi_A: X\to \mathbb{F}_2.$ In fact, for every function $\chi:X\to\mathbb{F}_2,$ we can define $A_\chi = \left\{x\in X \:|\:\chi\left(x\right) = 1\right\}.$ Combining this with the definition of the characteristic function, we have the identities

$A_{\chi_A} = A$ and $\chi_{A_\chi} = \chi.$

This actually provides the bijection between $\wp\left(X\right)$ and the functions $\chi$ that I mentioned immediately following the definition of the characteristic function.

Structurally, the class

$\mathbf{SSet}_X := \left\{\left(X, \chi\right) \:|\: \chi: X\to\mathbb{F}_2\right\}$

is very similar to the class

$\mathbf{MSet}_X := \left\{(\left(X, f\right) \:|\: f: X\to\mathbf{Card}\right\}$

of multisets over $X.$ Our ability to represent the set operations on $\mathbf{SSet}_X$ by well-defined operations in the ordered field $\mathbb{F}_2$ suggests that we look at some of the well-defined operations on $\mathbf{Card}$ as a basis for formally defining the multiset operations on $\mathbf{MSet}_X.$

One operation on $\mathbf{Card}$ that we have already used is the cardinal sum $\sum_{i\in I}\kappa_i,$ which is defined for every set of cardinals $\left\{\kappa_i\right\}_{i\in I}.$

Definition

Let $\mathcal{M} = \left\{\mathcal{M}_i = \left(X, f_i\right)\right\}_{i\in I}$ be an indexed family of multisets. The multiset sum of $\mathcal{M}$ is the multiset

$\biguplus_{i\in I} \mathcal{M}_i := \left(X, \sum_{i\in I}f_i\right).$

Since $\mathrm{dom}\left(f_i\right) = X$ for all $i\in I,$ the cardinal sum on the right is well-defined and the multiset sum does indeed give us another multiset over $X.$ For a pair of multisets, we will generally write the multiset sum in infix notation as $\mathcal{M}_1 \uplus \mathcal{M}_2.$

As the notation suggests, the multiset sum is related to the union operation on sets. Multiset sums, like the union of sets are both commutative and associative—these follow directly from the corresponding properties of the cardinal sum. Cardinal sums are also monotonic nondecreasing; for any cardinal sum $\kappa = \sum_{i\in I} {\kappa_i}$ we have $\kappa_i \leq \kappa$ for all $i\in I.$ One consequence of this is that a cardinal sum is 0 if and only if all of its summands are 0, and so we have the following relationship between support, multiset sums, and set unions:

$\mathrm{support}\left(\biguplus_{i\in I} {\mathcal{M}_i}\right) = \bigcup_{i\in I} {\mathrm{support}\left(\mathcal{M}_i\right)}.$

The monotonicity of cardinal sums also ensures that the multiset sum of a family of multisets contains each of the multisets in the family:

$\mathcal{M} = \biguplus_{i\in I}{\mathcal{M}_i}\:\Rightarrow\:\left(\forall i\in I\right)\left(\mathcal{M}_i\subseteq\mathcal{M}\right).$

Here’s a way in which the multiset sum differs from the union of sets. Set union is idempotent—for any set $X,$ we have $X\cup X = X.$ The multiset sum, on the other hand, is not idempotent, as we can see by considering the sum

$\left\{a\right\}\uplus\left\{a\right\} = \left\{a, a\right\} \neq \left\{a\right\}.$

This last example also shows that the class of multisets with all multiplicities equal to 1 is not closed under multiset sums.