Character of Pseudo-Tree Algebras

Abstract

For any Boolean algebra A, the character of A is χ(A)= min{κ: every ultrafilter on A can be generated by at most κ elements}. A pseudo-tree is a partial order (T,≤ _T ) such that for every t in T, the set {s∈T:s≤ _T t} is a linear order. The pseudo-tree algebra on T, denoted Treealg T, is the subalgebra of $\mathcal {P}(T)$ generated by the cones {s∈T:s≥ _T t}, for t in T. Ultrafilters on a pseudo-tree algebra Treealg T are in one-to-one correspondence with initial chains of T whenever T has a least element. We give an explicit description of a generating set of the ultrafilter corresponding to a given initial chain, and we use this to describe the character of Treealg T in terms of the structure of initial chains of T.