On Multiplicative Semigroups of Residue Classes

The set of residue classes, modulo any positive integer, is commutative and associative under the operation of multiplication. The author made the conjecture: For each finite commutative semigroup, S, there exists a positive integer, n, such that S is isomorphic with a subsemigroup of the multiplicative semigroup of residue classes (mod n). (A semigroup is a set closed with respect to a single-valued associative binary operation.) A counter-example to the above follows: Let S be the set, {z, a, b, c}, with all products defined to be z except bc = cb — a. Assume that there exists a positive integer, n, and four distinct residue classes, z, a, b, c (mod n) forming a system under multiplication isomorphic with S. Form the residue classes, 0=3 —2, a'=a — z, b' — b—z, c' — c — z. For x and y any ordered pair of z, a, 0, c, we have ix — z)iy—z)=xy~z — z+z = xy — z. Thus, 0, a', b', c' form a multiplicative semigroup isomorphic with S, where 0 is the zero residue class. The following must hold: o'2 and c'2 are each the zero residue class (mod n), but b'c' is a nonzero residue class; i.e., w|/32, ra|72, n\fiy, where /3 and 7 are integers in the residue classes 0' and c' (mod n) respectively. Since m|/32 and «|72, it follows that n2\$2y2, and n\(3y. The desired contradiction has been obtained. We begin the main result of this paper with a Definition. For each prime-power, pk, and each positive integer, m, we define a basic semigroup, aipk, m), with generators x and y, identity 1, annihilator 0, and commutative multiplication defined by: