  An interval algebra is a Boolean algebra which is isomorphic to the algebra
of finite unions of half-open intervals, of a linearly ordered set. An interval
algebra is hereditary if every subalgebra is an interval algebra. We answer a
question of M. Bekkali and S. Todor\v{c}evi\'c, by showing that it is
consistent that every $\sigma$-centered interval algebra of size $\mathfrak{b}$
is hereditary. We also show that there is, in ZFC, an hereditary interval
algebra of cardinality $\aleph_1.$
