Definition:Ordinal Multiplication

Definition

Let $x$ and $y$ be ordinals.

The operation of ordinal multiplication $x \times y$ is defined using transfinite recursion as follows:

Base Case

 $\ds x \times \O$ $=$ $\ds \O$ if $y = \O$

Inductive Case

 $\ds x \times z^+$ $=$ $\ds \paren {x \times z} + x$ if $y$ is the successor of some ordinal $z$

Limit Case

 $\ds x \times y$ $=$ $\ds \bigcup_{z \mathop \in y} \paren {x \times z}$ if $y$ is a limit ordinal

Also denoted as

The operation $x \times y$ is also seen denoted as $x \cdot y$ or, commonly, as $x y$.