B. Jónsson has asked : Is there an algebra of power a with no proper subalgebra of power a? H. J. Keisler and F. Rowbottom proved that the answer is affirmative for every a, provided Gödel's constructibility axiom holds [1]. The main aim of this paper is to prove, using the generalized continuum hypothesis (G .C .H. in what follows), that the answer is affirmative for each non limit cardinal (s...

If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ-strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to cons...

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ+ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which A = ∅. The first of these cont...