sentences:  The natural preorder on formula_17 is given by formula_22 if formula_23 or dually formula_24.While formula_25 and formula_26 agree on lattices, formula_25 properly refines formula_26 in the noncommutative case.The induced natural equivalence formula_29 is defined by formula_30 if formula_31, that is, formula_32
  and formula_33 or dually, formula_24 and formula_35.