Before revision:  We establish a variety of new results , including structural results for surplus-invariant acceptance sets and powerful dual representations for surplus-invariant risk measures .
After revision:  We establish a variety of new results , including structural results for surplus-invariant acceptance sets . We illustrate the power of the lattice approach by specifying our results to model spaces with a dominating probability, including Orlicz spaces, as well as to robust model spaces without a dominating probability, where the standard topological techniques and exhaustion arguments cannot be applied .