Chains and antichains in interval algebras

Abstract

Abstract Let κ be a regular cardinal, and let B be a subalgebra of an interval algebra of size κ . The existence of a chain or an antichain of size κ in ℬ is due to M. Rubin (see [7]). We show that if the density of B is countable, then the same conclusion holds without this assumption on κ . Next we also show that this is the best possible result by showing that it is consistent with 2 ℵ 0 = ℵ ω 1 that there is a boolean algebra B of size ℵ ω 1 such that length( B ) = ℵ ω 1 is not attained and the incomparability of B is less than ℵ ω 1 . Notice that B is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see. e.g., [1] and [2].