Boolean algebras with few endomorphisms

Using diamond for ml we construct a Boolean algebra in K}l, whose only endomorphisms are those definable using finitely many elements and ultrafilters. We also generalize Rubin's construction to higher cardinals. 0. Introduction. The aim we state in the name of the paper can be interpreted in two ways: every endomorphism is "simply defined", or the number of endomorphisms is small. However for every ultrafilter F on a Boolean algebra (or, equivalently maximal ideal I = B F) we can define an endomorphism T: