J. Donald Monk: Towers in Boolean algebras

Week 1, Friday June 19, 10:05-10:55

A tower in a BA A is a strictly increasing sequence of regular length of elements of A, with sum 1.
t_spect(A) = { |X|:X is a tower in A },
and t(A) = min (t_spect(A)).
Note that t(A) is not defined for every BA. We survey what is known about these functions in arbitrary BAs. Partly these results concern other cardinal functions generalized from the continuum cardinal case:
p(A)=min{ |X|: X sums to 1, but no finite subset of X does }
a(A)=min{ |X|: X is a partition of unity in A }
s(A)=min{ |X|: X splits A } (which means that for every nonzero a in A there is an x in X such that neither x nor -x is disjoint to a)
Results whose proofs are sketched:
(1) There is an atomless BA A such that p(A)<t(A).
(2) For uncountable regular kappa<lambda, there is an atomless BA A such that s(A)=t(A)<a(A).
(3) There is an atomless interval algebra A such that a(A)<t(A)
(4) If M is a nonempty set of regular cardinals, then there is an atomless BA A such that t_spect(A)=M.
(5) We give a characterization of those linear orders whose interval algebras have towers.
(6) A similar characterization is known for pseudo-tree algebras, and we sketch the special case of tree algebras.