On hereditarily rigid algebras (Q1083463)

An algebra A is said to be hereditarily rigid if every subalgebra of A has only one endomorphism. Let t be a similarity type; denote by R(t) the least cardinal number (if it exists) such that no t-algebra of cardinality \(\geq R(t)\) is hereditarily rigid. It is proved that if t is infinitary then R(t) does not exist; if t is unary then R(t) exists, and good bounds for this cardinal are found. If t is finitary and non-unary then R(t), if it exists, is strongly inaccessible. It is consistent with ZFC to assume one of the following possibilities: for every finitary non- unary t, R(t) does not exist; for every finitary t, R(t) exists; there are two finitary non-unary types t,t' such that R(t) does exist and R(t') does not exist.