Extensions of Certain Partial Automorphisms of L∗(V∞)

Abstract

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