The Weak Axiom of Revealed Preference (WARP) and distributability?

Question

If $(\mathcal{B}, C(\cdot))$ is a choice structure where $\mathcal{B}$ includes all non-empty subsets of consumption bundles $X$, i.e., $C(B) \neq \varnothing$ for all sets $B \in \mathcal{B}$, and the choice rule $C(\cdot)$ is distributive if, for any two sets $B$ and $B^{\prime}$ in $\mathcal{B}$ $$ C(B) \cap C\left(B^{\prime}\right) \neq \varnothing \text { implies that } C(B) \cap C\left(B^{\prime}\right)=C\left(B \cap B^{\prime}\right) $$ then, how can I develop a counterexample showing that, if the choice rule $C(\cdot)$ is distributive, then the choice structure $(\mathcal{B}, C(\cdot))$ will not necessarily satisfy the weak axiom of revealed preference?

Answer 1

Consider $X=\{x,y,z\}$, and the following choice rule:

$C(\{a\})=\{a\}, \ \forall a\in X$

$C(\{x,y\})=\{x\}$, $C(\{y,z\})=\{y\}$, $C(\{x,z\})=\{z\}$,

$C(\{x,y,z\})=\{x\}$

Clearly, the rule is distributive, but it violates WARP.