additive inverse

Since S is antinegative, only 0 has an additive inverse. Let us suppose that ([a.sub.1], [b.sub.1]) [member of] [S.sup.2] has an additive inverse, so there exists ([a.sub.2], [b.sub.2]) [member of] [S.sup.2], such that ([a.sub.1],[b.sub.1]) [direct sum] ([a.sub.2], [b.sub.2]) = (0, 0).

Semirings satisfy all properties of unital rings except the existence of additive inverses. Vandiver introduced the concept of semiring in [6], in connection with the axiomatization of the arithmetic of the natural numbers.

Rahn observes that if the nonzero intervals are paired as additive inverses (mod 7) to form the three interval classes (1, 6), (2, 5), (3, 4), then each of the two cycles .

For any odd prime p, let us define the set of interval classes (mod p) [ICS.sup.p] = {(1, p - 1), (2, p - 2), ...,((p - 1) / 2, p - (p - 1) / 2)} to be the reduced system of residues (mod p) in pairs of additive inverses (mod p), or, in musical terms, the set of interval classes excluding interval 0.