For the comparators I think that what he means is that for any given set X the set of comparators on X forms a semigroup where the composition is "compare using

For the comparators I think that what he means is that for any given set X the set of comparators on X forms a semigroup where the composition is "compare using the first comparator and then the second". Of course this too is a monoid with the identity being the comparator that says all things are equal.