  Let $G$ be a graph and suppose we are given, for each $v \in V(G)$, a strict
ordering of the neighbors of $v$. A set of matchings ${\cal M}$ of $G$ is
called internally stable if there are no matchings $M,M' \in {\cal M}$ such
that an edge of $M$ blocks $M'$. The sets of stable (\`a la Gale and Shapley)
matchings and of von Neumann-Morgenstern stable matchings are examples of
internally stable sets of matching. In this paper, we study, in both the
marriage and the roommate case, inclusionwise maximal internally stable sets of
matchings. We call those sets internally closed. By building on known and newly
developed algebraic structures associated to sets of matchings, we investigate
the complexity of deciding if a set of matchings is internally closed or von
Neumann-Morgenstern stable, and of finding sets with those properties.
