A starrable lattice is one with a cancellative semigroup structure satisfying (x ∨ y)(x ∧ y) = xy. If the cancellative semigroup is a group, then we say that the lattice is fully starrable. In this paper, it is proved that distributivity is a strict generalization of starrability. We also show that a lattice (X, ≤) is distributive if and only if there is an abelian group (G, +) and an injection f : X → G such that f (x) + f (y) = f(x ∨ y) + f (x ∧ y) for all x, y ∈ X , while it is fully starrable if and only if there is an abelian group (G, +) and a bijection f : X → G such that f (x) + f(y) = f (x ∨ y) + f (x ∧ y), for all x, y ∈ X.
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