An atomless interval Boolean algebra \(A\) such that \(\mathfrak {a}(A) <\mathfrak {t}(A)\). (Q1771931)

Let \(A\) be a Boolean algebra. A partition of unity in \(A\) is a collection of pairwise disjoint nonzero elements of \(A\) with sum equal to \(1\). \(\mathfrak {a}(A)\) denotes the smallest cardinality of an infinite partition of unity of \(A\). A tower in \(A\) is a subset \(X\) of \(A\) well-ordered by the induced order such that \(1\not \in X\) and \(\sum X=1\). \(\mathfrak {t}(A)\) denotes the smallest cardinality of a tower of \(A\). If \(L\) is a linearly ordered set with least element, the interval Boolean algebra is the algebra of subsets of \(L\) generated by the half-open intervals \([a,b)\). The author proves the existence of an atomless interval Boolean algebra \(A\) with \(\mathfrak {a}(A)<\mathfrak {t}(A)\).