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We compute the lattice operations for the (pairwise) stable set in two-sided matching markets where only substitutability on agents' choice functions is imposed. To do this, we use Tarski operators defined on the lattices of worker-quasi-stable and firm-quasi-stable matchings. These operators resemble lay-off and vacancy chain dynamics, respectively. First, we compute the lattice operations in the many-to-one model. Then, we extend these operations to a many-to-many model with substitutable choice functions on one side and responsive preferences on the other, via a morphism that relates many-to-one with many-to-many matchings in a natural way. Finally, we present the lattice operations in the many-to-many model with substitutable choice functions on both sides.
Bonifacio et al. (Tue,) studied this question.
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