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properties of ordinal arithmetic
Let On be the class of ordinals, and $\alpha,\beta,\gamma,\delta\in\textbf{On}$. Then the following properties are satisfied:
1. 2. (associativity of addition): $\alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$
3. (multiplicative identity): $\alpha\cdot 1=1\cdot\alpha=\alpha$
4. (multiplicative zero): $\alpha\cdot 0=0\cdot\alpha=0$
5. (associativity of multiplication): $\alpha\cdot(\beta\cdot\gamma)=(\alpha\cdot\beta)\cdot\gamma$
6. (left distributivity): $\alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma$
7. (existence and uniqueness of subtraction): if $\alpha\leq\beta$, then there is a unique $\gamma$ such that $\alpha+\gamma=\beta$
8.
Conspicuously absent from the above list of properties are the commutativity laws, as well as right distributivity of multiplication over addition. Below are some simple counterexamples:

$\omega+1\neq 1+\omega=\omega$, for the former has a top element and the latter does not.

$\omega\cdot 2\neq 2\cdot\omega$, for the former is $\omega+\omega$, which consists an element $\alpha$ such that $\beta<\alpha$ for all $\beta<\omega$, and the latter is $2\cdot\sup\{n\mid n<\omega\}=\sup\{2\cdot n\mid n<\omega\}=\sup\{n\mid n<\omega\}$, which is just $\omega$, and which does not consist such an element $\alpha$

$(1+1)\cdot\omega\neq 1\cdot\omega+1\cdot\omega$, for the former is $2\cdot\omega$ and the latter is $\omega\cdot 2$, and the rest of the argument follows from the previous counterexample.
All of the properties above can be proved using transfinite induction. For a proof of the first property, please see this link.
For properties of the arithmetic regarding exponentiation of ordinals, please refer to this link.
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