  In this work, we investigate the interplay between dissipation and
symmetry-protected topological order. We considered the one-dimensional spin-1
Affleck-Kennedy-Lieb-Tasaki model interacting with an environment where the
dissipative dynamics are described by the Lindladian master equation. The
Markovian dynamics is solved by the implementation of a tensor network
algorithm for mixed states in the thermodynamic limit. We observe that, for
time-reversal symmetric dissipation, the resulting steady state has topological
signatures even if being a mixed state. This is seen in finite string-order
parameters as well as in the degeneracy pattern of singular values in the
tensor network decomposition of the reduced density matrix. We also show that
such features do not appear for non-symmetric dissipation. Our work opens the
way toward a generalized and more practical definition of symmetry-protected
topological order for mixed states induced by dissipation.
