  Equality of the second order arithmetic means of two principal ideals does
not imply equality of their first order arithmetic means (second order equality
cancellation). We provide fairly broad sufficient conditions on one of the
principal ideals for this implication to hold true. We present also sufficient
conditions for second order inclusion cancellations. These conditions are
formulated in terms of the growth properties of the ratio of regularity
sequence associated to the sequence of s-number of a generator of the principal
ideal. These results are then extended to general ideals.
