  We address the question of the classification of gapped ground states in one
dimension that cannot be distinguished by a local order parameter. We introduce
a family of quantum spin systems on the one-dimensional chain that have a
unique gapped ground state in the thermodynamic limit that is a simple product
state but which on the left and right half-infinite chains, have additional
zero energy edge states. The models, which we call Product Vacua with Boundary
States (PVBS), form phases that depend only on two integers corresponding to
the number of edge states at each boundary. They can serve as representatives
of equivalence classes of such gapped ground states phases and we show how the
AKLT model and its $SO(2J+1)$-invariant generalizations fit into this
classification.
