Lattice-ordered injective hulls
Stuart A. Steinberg · Transactions of the American Mathematical Society · 1972
It is well known that the injective hull of a lattice-ordered group ( l l -group) M M can be given a lattice order in a unique way so that it becomes an l l -group extension of M M . This is not the case for an arbitrary f f -module over a partially ordered ring (po-ring). The fact that it is the case for any l l -group is used extensively to get deep theorems in the theory of l l -groups. For instance, it is used in the proof of the Hahn-embedding theorem and in the characterization of ℵ a {\aleph _a} -injective l l -groups. In this paper we give a necessary and sufficient condition on the injective hull of a torsion-free f f -module M M (over a directed essentially positive po-ring) for it to be made into an f f -module extension of M M (in a unique way). An f f -module is called an i − f i - f -module if its injective hull can be made into an f f -module extension. The class of torsion-free i − f i - f -modules is closed under the formation of products, sums, and Hahn products of strict