  Motzkin spin chains are frustration-free models whose ground-state is a
combination of Motzkin paths. The weight of such path contributions can be
controlled by a deformation parameter t. As a function of the latter these
models, beside the formation of domain wall structures, exhibit a
Berezinskii-Kosterlitz-Thouless phase transition for t=1 and gapped Haldane
topological orders with constant decay of the string order parameters for t <
1. By means of numerical calculations we show that the topological properties
of the Haldane phases depend on the spin value. This allows to classify
different kinds of hidden antiferromagnetic Haldane gapped regimes associated
to nontrivial features like symmetry-protected topological order. Our results
from one side allow to clarify the physical properties of Motzkin
frustration-free chains and from the other suggest them as a new interesting
and paradigmatic class of local spin Hamiltonians.
