  The arithmetic mean is the mean for addition and the geometric mean is that
for multiplication. Then what kind of binary operation is associated with the
arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to
construct an arithmetic operation such that AGM is the mean for this operation,
it can be regarded as an intermediate operation between addition and
multiplication in view of the meaning of AGM. In this paper such an operation
is introduced and several of its algebraic properties are proved.
