TITLE: Commutative addition on the ordinals
QUESTION [9 upvotes]: It is well known that ordinal addition is not commutative (for example $\omega+1\neq 1+\omega$), but it is associative. My question regards a new kind of addition defined as:
$$a\oplus b = \text{max}\{a+b,b+a\}$$
This addition is obviously commutative, but is it associative? I can't find a counterexample, but I also can't prove that it is.
Thank you very much.

REPLY [9 votes]: This is not associative.
$$\begin{align}a\oplus(b\oplus c)&=\max\{a+(b\oplus c),(b\oplus c)+a\}\\
&=\max\{a+\max\{b+c,c+b\},\max\{b+c,c+b\}+a\}\\
&=\max\{a+b+c,a+c+b,b+c+a,c+b+a\} \end{align}$$
and
$$\begin{align}(a\oplus b)\oplus c&=\max\{(a\oplus b)+c,c+(a\oplus b)\}\\
&=\max\{\max\{a+b,b+a\}+c,c+\max\{a+b,b+a\}\}\\
&=\max\{a+b+c,b+a+c,c+a+b,c+b+a\} \end{align}$$
Thus for example
$$ \omega^2\oplus(1\oplus\omega)=\omega^2+\omega+1$$
and 
$$ (\omega^2\oplus1)\oplus\omega=\omega^2+\omega$$