Modules lattice isomorphic to linearly compact modules

We study modules that are lattice isomorphic to linearly compact modules (in the discrete topology). In particular, we establish the equivalence of the following properties of a module $M$: 1) $M$ satisfies the Grothendieck property \textrm{AB$5^*$} and all its submodules are Goldie finite-dimensional; 2) $M$ is lattice isomorphic to a linearly compact module; 3) $M$ is lattice antiisomorphic to a linearly compact module. We show that a linearly compact module cannot be determined in terms of the lattice of its submodules.