  Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that appear in imprecise-probabilistic
decision making. We provide these choice functions with a clear interpretation
in terms of desirability, use this interpretation to derive a set of basic
coherence axioms, and show that this notion of coherence leads to a
representation in terms of sets of strict preference orders. By imposing
additional properties such as totality, the mixing property and Archimedeanity,
we obtain representation in terms of sets of strict total orders, lexicographic
probability systems, coherent lower previsions or linear previsions.
