Before revision: For any matching market where the set of stable matchings has a lattice structure, if the binary operations ( l.u.b. and g.l.b. ) are computed via a "pointing function", then the set of random matchings (lotteries of stable matchings) has a lattice structure .
After revision: For any matching market where the set of stable matchings has a lattice structure, if the binary operations ( l.u.b. and g.l.b. (greatest lower bound) for both sides of the matching market, and we prove the that with these binary operations the set of random matchings (lotteries of stable matchings) has a lattice structure .