On hereditary interval algebras

We show that each hereditary interval algebra has a countable density and not conversely. Moreover, we show that, for an interval algebra, having countable density and being subal-gebra of the interval algebra over the real line are equivalent statements. 1. Introduction. Boolean algebras that are generated by subchains, that is, subsets that are linearly ordered under the Boolean partial order, were introduced in 1939 by Mostowski and Tarski [7] and have been extensively studied since then. Nowadays they are called interval algebras. All basic facts about these algebras can be found in [6, Section 15]. We remark, at this stage, that a subalgebra of an interval algebra need not be an interval algebra. For instance, one can consider the algebra of finite and co-finite subsets of the first uncountable cardinal. This leads us to the study of hereditary interval algebras, that is, those algebras of which any subalgebra is an interval algebra. The main concern of this note is to shed more light on these algebras. This note is organized as follows. Section 2 deals with definitions. Section 3 is a presentation of the main theorem. In Section 4, some consequences of this theorem are given.