Stiff algebras (Q1905984)

The article is concerned with the consistency of the existence of atomic Boolean algebras which are close to rigid. A BA is Hopfian if every onto endomorphism is 1-1; dual Hopfian if every 1-1 endomorphism is onto; almost rigd if no automorphism moves infinitely many atoms; stiff if it satisfies all of these properties; tree-like if mod finite it is a tree algebra; almost free if it has a generating set which is a free generating set in the quotient mod finite. There is a more complicated notion of weakly endo-rigid which enters into one result mentioned below. The main results are: (1) There is a consistent example of a stiff almost free atomic BA of size less than \(2^\omega\); (2) There is a consistent tree-like stiff atomic BA; (3) No tree-like algebra of size less than \(2^\omega\) can be either Hopfian, dual Hopfian, or almost rigid; (4) No almost free or tree-like atomic BA is weakly endo-rigid.