The Length of Chains in Modular Algebraic Lattices

Abstract

We show that, for a large class of countable order types α, a modular algebraic lattice L contains no chain of type α if and only if K(L), the join-semilattice of compact elements of L, contains neither a chain of type α nor a join-subsemilattice isomorphic to [ω] < ω , the set of finite subsets of ω ordered by inclusion. We give a description of the indivisible members of this class. It includes the order types ω * of the chain of negative integers and η of the chain of rational numbers.