Compatible additive permutations of finite integral bases

Let be an additive permutation of a finite integral base. It is shown that ifB is symmetric, then there is a unique additive permutation ofB which is compatible with in the sense that ^–1 is also an additive permutation; and that, further, ifB is asymmetric, then there is no additive permutation ofB which is compatible with. Thus, in the symmetric case, there are no additively compatible sets (of permutations) forB of size greater than 3. This contrasts with the situation for completely compatible sets (equivalently, additive sequences of permutations) where for certainB compatible sets of size (resp. length) 4 or less are known, but where nothing is known of sets of greater size (resp. length). It is also noted how this result restricts the possibility of a useful multiplication theorem for the additive analogue of perfect systems of difference sets and graceful graphs.