  Information theoretical inequalities have strong ties with polymatroids and
their representability. A polymatroid is entropic if its rank function is given
by the Shannon entropy of the subsets of some discrete random variables. The
book is a special iterated adhesive extension of a polymatroid with the
property that entropic polymatroids have $n$-page book extensions over an
arbitrary spine. We prove that every polymatroid has an $n$-page book extension
over a single element and over an all-but-one-element spine. Consequently, for
polymatroids on four elements, only book extensions over a two-element spine
should be considered. F. Mat\'{u}\v{s} proved that the Zhang-Yeung inequalities
characterize polymatroids on four elements which have such a 2-page book
extension. The $n$-page book inequalities, defined in this paper, are
conjectured to characterize polymatroids on four elements which have $n$-page
book extensions over a two-element spine. We prove that the condition is
necessary; consequently every book inequality is an information inequality on
four random variables. Using computer-aided multiobjective optimization, the
sufficiency of the condition is verified up to 9-page book extensions.
