Cardinal functions on initial chain algebras on pseudotrees

Initial chain algebras on pseudotrees generalize the notion of an interval algebra on a linear order. Many relationships which hold between the various cardinal functions on interval algebras also hold for initial chain algebras. In particular, for initial chain algebras on pseudotrees, depth equals tightness, spread equals hereditary Lindelof degree, irredundance equals the cardinality of the algebra, and incomparability equals hereditary cofinality. For interval algebras, Rubin showed that any subalgebra of regular uncountable cardinality kappa contains either a chain of size kappa or a pairwise incomparable family of size kappa. This result holds for initial chain algebras as well.