On the endomorphism monoids of Boolean algebras and distributive lattices

This work is concerned with some questions which arise in the study of endomorphism monoids of Boolean algebras and finite distributive lattices. It is a well known fact that the automorphism group does not characterize a Boolean algebra. New examples to verify the above fact are obtained by considering the finite confinite Boolean algebra on an infinite set X, and its generalizations, we obtain new examples of non-isomorphic Boolean algebras in which the automorphism groups are isomomorphic. It is shown that there exist an infinite number of non-isomorphic Boolean algebras with automorphism groups anti-isomorphic to S(X), the symmetric group of transformations on a set X. C. J. Maxson has shown that for the Boolean algebra 2 [superscript X], the monoid of complete endomorphisms is anti-isomorphic to T(X), the monoid of transformations on X. This motivates the problem of finding Boolean algebras for which every endomorphism is complete. It is shown here that if U is a complete (atomic) Boolean algebra then every endomorphism of U is complete if and only if U is finite