Review: Associativity, commutativity, and idempotence
A binary operation on a set is a function . It is usually written using an infix notation rather than a functional notation such as .
The operation is associative if and only if for all . It is commutative if and only if for all . It is idempotent if and only if for all .
We’ve defined four operations on families of multisets where for all : sums, products, intersections, and unions. As I’ve already commented in a couple of places, we can restrict our attention to families where . This provides us with four binary operations on . Today’s post collects together the properties of these binary operations.
Proposition 1: The operations , , , and on are all associative and commutative.
Proof:
Let . The first four parts follow directly from the associativity and commutativity of and as binary operations on .
Associativity of .
Commutativity of .
Associativity of .
Commutativity of .
The remaining parts follow from the following facts:
- Let be partially ordered sets. Then and whenever the suprema and infima exist.
- The supremum and infimum of every set of cardinals exist.
Associativity of .
Commutativity of .
Associativity of .
Commutativity of .
Proposition 2: The operations and on are idempotent.
Proof:
Let .
Idempotence of .
Idempotence of .
The other two operations—sum and product—are not idempotent. One counterexample is the multiset , for which we have
and
.
Copyright © 2008 Michael L. McCliment.