  Partial symmetries are described by generalized group structures known as
symmetric inverse semigroups. We use the algebras arising from these structures
to realize supersymmetry in (0+1) dimensions and to build many-body quantum
systems on a chain. This construction consists in associating appropriate
supercharges to chain sites, in analogy to what is done in spin chains. For
simple enough choices of supercharges, we show that the resulting states have a
finite non-zero Witten index, which is invariant under perturbations, therefore
defining supersymmetric phases of matter protected by the index. The
Hamiltonians we obtain are integrable and display a spectrum containing both
product and entangled states. By introducing disorder and studying the
out-of-time-ordered correlators (OTOC), we find that these systems are in the
many-body localized phase and do not thermalize. Finally, we reformulate a
theorem relating the growth of the second Renyi entropy to the OTOC on a
thermal state in terms of partial symmetries.
