An interval algebra is a Boolean algebra \(B\) which is isomorphic to the algebra of finite unions of half-open intervals, of a linearly ordered set. The interval algebra \(B\) is hereditary if all its subalgebras are interval algebras. M. Bekkali and S. Todorčević [Algebra Univers. 73, No. 1, 87–95 (2015; Zbl 1338.06013)] showed that every hereditary interval algebra is \(\sigma\)-centered. But not all \(\sigma\)-centered interval algebras are hereditary.
Here, the authors investigate the natural cardinal invariant \(\mathfrak{h} \mathfrak{i} \mathfrak{a}\), which denotes the minimal cardinality of a non-hereditary interval \(\sigma\)-centered algebra. It is known that \(\mathfrak{h} \mathfrak{i} \mathfrak{a} \geq \mathfrak{b}\).
In this paper, it is shown that for every uncountable regular cardinal \(\kappa\), it is consistent that \(\mathfrak{b} \leq \mathfrak{h} \mathfrak{i} \mathfrak{a} = \kappa\) and \(\mathfrak{b} = \aleph_1\). The authors answer a question of Bekkali and Todorčević [loc. cit.] by showing that it is consistent that every \(\sigma\)-centered interval algebra of size \(\mathfrak{b}\) is hereditary. They introduce a mild strengthening \(\mathfrak{b}_2\) of \(\mathfrak{b}\) and show that \(\mathfrak{b} \leq \mathfrak{h} \mathfrak{i} \mathfrak{a} \leq \mathfrak{b}_2\) and they complete their results by showing that in ZFC, there is a hereditary interval algebra of cardinality \(\aleph_1\).