A class of multipartner matching markets with a strong lattice structure

For a two-sided multipartner matching model where agents are given by path-independent choice functions and no quota restrictions, Blair [7] had shown that stable matchings always exist and form a lattice. However, the lattice operations were not simple and not distributive. Recently Alkan [3] showed that if one introduces quotas together with a monotonicity condition then the set of stable matchings is a distributive lattice under a natural definition of supremum and infimum for matchings. In this study we show that the quota restriction can be removed and replaced by a more general condition named cardinal monotonicity and all the structural properties derived in [3] still hold. In particular, although there are no exogenous quotas in the model there is endogenously a sort of quota; more precisely, each agent has the same number of partners in every stable matching. Stable matchings also have the polarity property (supremum with respect to one side is identical to infimum with respect to the other side) and a property we call {\it complementarity}.