$(P(X), C_R)$ may be a choice structure even if $R$ is not a rational relation.

Question

Would you please give me an example to show that $(P(X), C_R)$ may be a choice structure even if $R$ is not rational (i.e., complete and transitive).

Clarification:

• For any nonempty set $X$, let $P(X)$ denote the set of all nonempty subsets of X.
• For any nonempty subset $B$ of $P(X)$, a function $c: B \rightarrow P(X)$ is called a choice function iff $c(A) \subset A$ for all $A \in B$. The pair $(B, c)$ is called a choice structure.
• For any binary relation $R$ on $X$, define the function $C_R : P(X) \rightarrow P(X) \cup \{\emptyset\}$ as follows: $$C_R (A) = \{x \in A : ( \forall y \in A ) ( xRy ) \}.$$

Answer 1

No non-complete relation $R$ can yield a choice structure in this way. Suppose neither $aRb$ nor $bRa$ holds. Then the map $C_R$ is not a choice function: $C_R(\{a, b\})=\emptyset$.

On the other hand, note that $R$ being rational isn't even enough: if $R$ is the usual ordering on $\mathbb{R}$, then $C_R(\mathbb{Z})=\emptyset$.