 \(\mathbf{C}^L\) is a choice function if and only if \(\succcurlyeq\) is acyclical. It will then satisfy properties \(\alpha\) and \(\gamma\). Furthermore, \(\mathbf{C}^L\) satisfies property \(\beta\) if and only if \(\succcurlyeq\) is transitive and complete (Herzberger 1973).