  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.
