What’s between addition and multiplication? Since the space between addition or multiplication of reals is continuous (ie, the space between 2+3=5 and 2*3=6 is dense), there ought to be *something* there that can be characterized. An arithmetic means like ((a+b)+(ab))/2 or a geometric means like sqrt((ab)(a+b)) are just hacks as far as I’m concerned, since they’re amalgams of operations that are not fundamentally ‘between’ between addition and multiplication.

Here’s a more formal restatement this question: if f(0,x,y)=x+y and f(1,x,y)=x*y, find a function f(1/2,x,y), where for any x and y, if f(0,x,y)<f(1,x,y) then f(0,x,y)<f(1/2,x,y)<f(1,x,y) and vice versa.

I found one promising lead using partial iterates of exponential functions. Consider h(x,y)=x+y and g(x,y)=xy=exp(log(x)+log(y)). Here, multiplication *is* addition in exponential space. Now, define f(a,x,y) = exp[a]^(log[a](x) + log[a](y)), where notation exp[a](x) means an a-th iterated function of the exponential function exp(x) or e^x. For example, exp[3](x)=e^(e^(e^x)) or sin[2](x)=sin(sin(x)) or log[-1](x)=1-th inverse of log(x) function=e^x. Since the zeroth iterate of any function is the identity function Id(z)=z, h(0,x,y) = exp[0](log[0](x)+log[0](y)) = Id(Id(x)+Id(y)) = x+y = f(x,y). And, h(1,x,y) = exp[1](log[1](x)+log[1](y)) = exp(log(x)+log(y)) = exp(log(xy)) = xy = g(x,y). Then, we could think of interpolating between addition and multiplication by varying this parameter ‘a’ between 0 and 1.