We introduce weight-preserving sets of binary words. Any transformation that respects the row and column weights of a 0-1 matrix can be decomposed as a composition of two types of action on the matrix. We conjecture that weight-preserving sets perform only one type of action, permutations of rows and columns; i.e., weight-preserving sets are cwatsets.
Badger, Matthew, "On the Action of Weight-Preserving Sets" (2005). Mathematical Sciences Technical Reports (MSTR). 53.