Let G be an arbitrary lattice-ordered group. We define a topology on G, called the J \mathcal {J} -topology, which is a group and lattice topology for G and which is preserved by cardinal products. The J \mathcal {J} -topology is the interval topology on totally ordered groups and is discrete if and only if G is a lexico-sum of lexico-extensions of the integers. We derive necessary and sufficient conditions for the J \mathcal {J} -topology to be Hausdorff, and we investigate J \mathcal {J} -topology convergence.