An atomless interval Boolean algebra A such that αAtA

For any Boolean algebra A, a(A) is the smallest cardinality of an infinite partition of unity in A. A tower in a Boolean algebra A is a subset X of A well-ordered by the Boolean ordering, with 1 is not an element of X but with Sigma X = 1. t(A) is the smallest cardinality of a tower of A. Given a linearly ordered set L with first element, the interval algebra of L is the algebra of subsets of L generated by the half-open intervals [a, b). We prove that there is an atomless interval algebra A such that a(A) < t(A).