No Abelian Semigroup Operation Is Complete

This paper shows that there does not exist a finite abelian semigroup <S, > with order > 3 such that the semigroup operation is complete over S. Neither is {} complete with constants over S. J. C. Muzio has shown that over any finite space there exists a set of two abelian semigroup operations which is complete with constants over the space. Hence Muzio's result is best possible.