Title: A generalization of the injectivity condition for Projected Entangled Pair States,
Abstract: We introduce a family of tensor network states that we term semi-injective
Projected Entangled-Pair States (PEPS). They extend the class of injective PEPS
and include other states, like the ground states of the AKLT and the CZX models
in square lattices. We construct parent Hamiltonians for which semi-injective
PEPS are unique ground states. We also determine the necessary and sufficient
conditions for two tensors to generate the same family of such states in two
spatial dimensions. Using this result, we show that the third cohomology
labeling of Symmetry Protected Topological phases extends to semi-injective
PEPS.