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}.

Copyright © 2008 Michael L. McCliment.