Means Compatible with Semigroup Laws

Abstract

A binary mean operation m(x,y) is said to be compatible with a semigroup law *, if * satisfies the Gauss' functional equation m(x,y) * m(x,y) = x * y for all x, y. Thus the arithmetic mean is compatible with the group addition in the set of real numbers, while the geometric mean is compatible with the group multiplication in the set of all positive real numbers. Using one of Jacobi's theta functions, Tanimoto has constructed a novel binary operation * corresponding to the arithmetic-geometric mean agm(x,y) of Gauss. Tanimoto shows that it is only a loop operation, but not associative. A natural question is to ask if there exist a group law * compatible with arithmetic-geometric mean. In this paper we prove that there is no semigroup law compatible with agm and hence, in particular, no group law either. Among other things, this explains why Tanimoto's novel operation * using theta functions must be non-associative.