Pi Mu Epsilon
Most day to day calculations take place within the field of real numbers with the two binary operations of addition and multiplication. In this field, these two operations are definitionally independent of one another. However, if we approach binary operations from a different point of view, e.g. that of recursive formulae, we can develop multipli cation from addition by use of the concept of repeated addition. Along similar lines, we can develop exponentiation from multiplication by re peated multiplication. The next logical step would be to try to develop another binary operation based on repeated exponentiation.
Smolarski, D. C. (1975). Generalizing Binary Operations. Pi Mu Epsilon Journal, 6(2), 75–78. http://www.pme-math.org/journal/issues/PMEJ.Vol.6.No.2.pdf