Definition:Ordinal Multiplication

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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 $x y$.