Communications in Computer and Information Science, Volume 3019 (IPMU 2026), June 2026.
We study independence concepts for imprecise-probabilistic choice functions. Leveraging the idea that a choice function is represented by a collection of coherent sets of desirable gambles, we introduce representational independence as the requirement of having an independent representation. We derive the representational product, which is the smallest coherent choice function that marginalises to given choice functions and satisfies representational independence. By applying similar proof techniques, we also obtain the independent product for the existing S-independence. We compare these products, together with the existing epistemic product, and derive some of their properties. Finally, we discuss open questions concerning the independence notions.