Here are the two binary operations (I'm not sure how to format the tables here):
$*|a$ $b$ -------- $*|a$ $b$
$a|a$ $a$ -------- $a| a$ $b$
$b|a$ $b$ --------- $b| b$ $b$
Usually when I show something is associative, I have three elements, so I'm not sure how to go about showing associativity here.