Definition:Ordinal Multiplication

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Let $x$ and $y$ be ordinals.

The operation of ordinal multiplication $x \times y$ is defined using the Second Principle of 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 $x y$.