# Definition:Ordinal Multiplication

Jump to navigation Jump to search

## Definition

Let $x$ and $y$ be ordinals.

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

### Base Case

 $\displaystyle \left({x \times \varnothing}\right)$ $=$ $\displaystyle \varnothing$ if $y$ is equal to $\varnothing$

### Inductive Case

 $\displaystyle \left({x \times z^+}\right)$ $=$ $\displaystyle \left({x \times z}\right) + x$ if $y$ is the successor of some ordinal $z$

### Limit Case

 $\displaystyle \left({x \times y}\right)$ $=$ $\displaystyle \bigcup_{z \mathop \in y} \left({x \times z}\right)$ 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 $x y$.