\begin{definition}[Definition:Canonical Order]
Let $\On$ be the ordinal class.
Let $<$ be the ordinal ordering.
Let $\otimes^l$ be the lexicographic ordering on $\On \times \On$.
The '''canonical order''' on $\On \times \On$, denoted $R_0$, is defined as follows, for ordinals $\alpha, \beta, \gamma, \delta$:
::$\tuple {\alpha, \beta} \mathrel {R_0} \tuple {\gamma, \delta}$
:{{iff}}:
::$\map \max {\alpha, \beta} < \map \max {\gamma, \delta}$ or $\paren {\map \max {\alpha, \beta} = \map \max {\gamma, \delta} \land \tuple {\alpha, \beta} \otimes^l \tuple {\gamma, \delta} }$
\end{definition}