  How many endomorphisms does a Boolean algebra have? Can we find Boolean
algebras with as few endomorphisms as possible? Of course from any ultrafilter
of the Boolean algebra we can define an endomorphism, and we can combine
finitely many such endomorphisms in some reasonable ways. We prove that in any
cardinality lambda=lambda^ {aleph_0} there is a Boolean algebra with no other
endomorphisms. For this we use the so called "black boxes", but in a self
contained way. We comment on how necessary the restriction on the cardinal is.
