  Most comparisons of preferences are instances of single-crossing dominance.
We examine the lattice structure of single-crossing dominance, proving
characterisation, existence and uniqueness results for minimum upper bounds of
arbitrary sets of preferences. We apply these theorems to derive new
comparative statics theorems for collective choice and under analyst
uncertainty, to characterise a general 'maxmin' class of uncertainty-averse
preferences over Savage acts, and to revisit the tension between liberalism and
Pareto-efficiency in social choice.
