Separable discrete preferences

An ordering of multidimensional alternatives is separable on a set of dimensions if fixing values on the complementary dimensions always produces the same induced ordering. Most often, studies of separability assume continuous alternative spaces; as we show, separability has different properties when alternative spaces are discrete. For instance, two well-known theorems of Gorman-that common set operations preserve separability and that separable preferences are additive-fail for binary alternative spaces. Intersection is the only set operation that preserves separability. For binary alternative spaces, separability is equivalent to additivity if and only if there are four or fewer dimensions.