We compute the lattice operations for the (pairwise) stable set in many-to-many matching markets when only path-independence on agents' choice functions is imposed. To do this, we first show that the sets of firm-quasi-stable and worker-quasi-stable many-to-many matchings form lattices. Then, we construct Tarski operators on these lattices whose fixed points coincide with the set of stable matchings, and show that iterating these operators from suitable quasi-stable matchings yields the lattice operations in the stable set. These operators resemble lay-off and vacancy chain dynamics, respectively.

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