In an earlier post, What is known about the category of monads on Set? the following observation was made:
What's more, all but two monads on Set have the property that there exists an algebra with more than one element. One of the exceptions is the monad M with M(A)=1 for all sets A; it's the theory generated by a single constant e and the equation x=e. The other is the monad M with M(A)=1 for all nonempty sets A and M(0)=0; that's the theory generated by no operations and the equation x=y.
Does anyone know a proof (or a reference to a proof) of this fact--that these are the only two monads on sets having trivial algebras?