# Hereditary Interval Algebras and Cardinal Characteristics of the Continuum

## TL;DR

This paper explores the properties of hereditary interval algebras, demonstrating their consistency with certain set-theoretic assumptions and establishing their existence within ZFC, thereby advancing understanding of their structure and cardinal characteristics.

## Contribution

It proves the consistency of all σ-centered interval algebras of size  with being hereditary and constructs an hereditary interval algebra of size  within ZFC.

## Key findings

- Every -centered interval algebra of size  can be hereditary under certain set-theoretic assumptions.
- There exists an hereditary interval algebra of size  in ZFC.
- The paper answers a question posed by Bekkali and Todordevi
07.

## Abstract

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 $\aleph_1.$