$x\oplus y=y\oplus x$ for all $x$ and $y$

$x\oplus 0=|x|$ for all $x$

$x\oplus (-x)=0$ for all $x$

$x\oplus y\ge 0$ for all $x$ and $y$

$(x\oplus y)\oplus z=x\oplus (y\oplus z)$ for all $x,y,z$