Restricted choice functions with a fallback are exactly those definable by a linear order on the allowed sets, characterized by postulates (SS0)-(SS6) in the union-closed case and (SS0)-(SS4), (SS5E), (SS6E) in general.
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Axiomatics of Restricted Choices by Linear Orders of Sets with Minimum as Fallback
Restricted choice functions with a fallback are exactly those definable by a linear order on the allowed sets, characterized by postulates (SS0)-(SS6) in the union-closed case and (SS0)-(SS4), (SS5E), (SS6E) in general.