Extensions of Certain Partial Automorphisms of $\mathcal{L}^{\mathcal{ \ast }}(V_{\infty })$

The automorphisms of the lattice $\mathcal{L}(V_{\infty })$ have been completely characterized. However, the question about the number of automorphisms of the lattice $\mathcal{L}^{\mathcal{ \ast }}(V_{\infty })$ has been open for almost thirty years. We use some of our recent results about the structure of $\mathcal{L}^{\mathcal{ \ast }}(V_{\infty })$ to answer questions related to automorphisms of $\mathcal{L}^{\mathcal{ \ast }}(V_{\infty })$. We prove that any finite number of partial automorphisms of filters of closures of quasimaximal sets can be extended to an automorphism of $\mathcal{L}^{\mathcal{ \ast }}(V_{\infty })$. As a corollary we obtain that closures of quasimaximal sets of the same type are elements of the same orbit in $\mathcal{L}^{\mathcal{ \ast }}(V_{\infty })$.