Hilbert lattices with the extension property (Q1119956)

The author proves that there exist Hilbert lattices L with the property (extension) that each ortho-isomorphism \(\tau\) : [0,a]\(\to [0,b]\) between arbitrary interval sublattices of height at least 3 in the orthocomplemented lattice L can be extended to an ortho-automorphism of L. These examples of lattices with the extension property are of infinite height. On the level of sesquilinear spaces the extension property is of interest because this property implies orthomodularity of the underlying polarity lattice of orthoclosed subspaces. The author also constructs Hilbert lattices that do not have the extension property but do enjoy the cancellation property.