Before revision:  We define the l.u.b. ( least upper bound ) and theg.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 has two dual lattices.
After revision:  We define the l.u.b. ( least upper bound ) and the ( greatest lower bound ) for both sides of the matching market, and we prove the that with these binary operations the set of random matchings has two dual lattices.