  We introduce a general technique for decomposing monomial algebras which we
use to study the Lefschetz properties. We apply our technique to various
classes of algebras, including monomial almost complete intersections and
Gorenstein algebras. In particular, we prove that Gorenstein codimension three
algebras arising from numerical semigroups have the strong Lefschetz property.
We also study the reverse of the splitting operation -- a gluing operation --
which gives a way to construct monomial algebras with the Lefschetz properties.
