There's no need to think in terms of rings here ((+,*) and (|,&) together) - Monoids are precisely the most powerful algebraic object we need to consider here (

There's no need to think in terms of rings here ((+,*) and (|,&) together) - Monoids are precisely the most powerful algebraic object we need to consider here (only one binary operation at a time). All of these binary operations form a monoid.