These intermediate operations were topic already in this thread.

For me it seems important that the curve t |-> a [t] b is smooth (or better analytic) for fixed a and b.

I mean you define it on the interval [1,2], i.e. between addition and multiplication, and then you would continue it to the higher operations t>2 by

a [t+1] (b+1) = a [t] ( a [t+1] b )

And then it would be interesting whether the curve is (infinitely) differentiable at t=2, t=3, etc. Of course it would between the endpoints, i.e. on (2,3) and (3,4), etc.

For me it seems important that the curve t |-> a [t] b is smooth (or better analytic) for fixed a and b.

I mean you define it on the interval [1,2], i.e. between addition and multiplication, and then you would continue it to the higher operations t>2 by

a [t+1] (b+1) = a [t] ( a [t+1] b )

And then it would be interesting whether the curve is (infinitely) differentiable at t=2, t=3, etc. Of course it would between the endpoints, i.e. on (2,3) and (3,4), etc.