ABSTRACT
Perhaps, multiplicative systems would have come about from various sources. One such is the set N of positive integers where ordinary multiplication gives rise to a monoid (N, ·) with 1 as the unity element. George Cantor, the father of set theory, gave the example of all maps: Map (A,B) from a non-empty set A into a non-empty set B. The composition of maps denes a ‘multiplication’ which is associative. For composition of Map (A,B) and Map (B,C) where A,B,C are non-empty sets one gets Map (A,C). In particular, Map (A,A) satises the associative law of composition and the identity map iA : A→ A (onto A) serves as the multiplicative identity. As is well-known, a semigroup is a multiplicative system in which the associative law for multiplication holds. The evolution of groups has its roots in geometry and in analysis, and group-theoretic ideas were used around 1800. As (N, ·) is a semigroup, it is natural to consider functions from a semigroup, G into the eld C of complex numbers. In [11], J. Knopfmacher gives an interesting exposition of abstract analytic number theory wherein many of the results of classical number theory are generalized in a suitable context of semigroups satisfying certain axioms. The treatment is all the more remarkable when one gets an abstract analogue of the Prime Number Theorem.
