A lattice is called groupable provided it can be endowed with the structure of anl-group (lattice ordered group). The primary objective of this paper is to introduce an order theoretic property of groupable lattices which implies that all associatedl-groups are subdirect products of totally ordered groups. This is an analog to Iwasawa's well-known result which asserts that a conditionally completel-group is abelian. A secondary objective is to outline a general method for identifying classes ofl-groups determined by order theoretic properties.