1: The lattice structure of the subflattening row or column indices. Here, 1 is the 'special' state and 0 represents other states. For example if k is the special state, 101 corresponds to words ksk for s ̸ = k. (a) highlights the possible indices for the flattening, (b) the (2, 2)-subflattening and (c) the (1, 1)-subflattening. (d) corresponds to the nonfaithful representation which maps group elements to the 1 × 1 identity matrix. To visualise indexing in this way for a (3, 2)-subflattening we would draw the lattice (a) to represent the possible column labellings, and (b) for the possible row labellings.