  Area law violations for entanglement entropy in the form of a square root has
recently been studied for one-dimensional frustration-free quantum systems
based on the Motzkin walks and their variations. Here we consider a Motzkin
walk with a different Hilbert space on each step of the walk spanned by
elements of a {\it Symmetric Inverse Semigroup} with the direction of each step
governed by its algebraic structure. This change alters the number of paths
allowed in the Motzkin walk and introduces a ground state degeneracy sensitive
to boundary perturbations. We study the frustration-free spin chains based on
three symmetric inverse semigroups, $\cS^3_1$, $\cS^3_2$ and $\cS^2_1$. The
system based on $\cS^3_1$ and $\cS^3_2$ provide examples of quantum phase
transitions in one dimensions with the former exhibiting a transition between
the area law and a logarithmic violation of the area law and the latter
providing an example of transition from logarithmic scaling to a square root
scaling in the system size, mimicking a colored $\cS^3_1$ system. The system
with $\cS^2_1$ is much simpler and produces states that continue to obey the
area law.
