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When making simultaneous decisions, our preference for the outcomes on onesubset can depend on the outcomes on a disjoint subset. In referendumelections, this gives rise to the separability problem, where a voter mustpredict the outcome of one proposal when casting their vote on another. A set$S \subset [n]$ is separable for preference order $\succeq$ when our ranking ofoutcomes on $S$ is independent of outcomes on its complement $[n]-S$. Theadmissibility problem asks which characters $\mathcal{C} \subset\mathcal{P}([n])$ can arise as the collection of separable subsets for somepreference order. We introduce a linear algebraic technique to constructpreference orders with desired characters. Each vector in our $2^n$-dimensionalvoter basis induces a simple preference ordering with nice separabilityproperties. Given any collection $\mathcal{C} \subset \mathcal{P}([n])$ whosesubset lattice has a tree structure, we use the voter basis to construct apreference order with character $\mathcal{C}$.

The Voter Basis and the Admissibility of Tree Characters