Can we extend operations beyond positive integers?

The first operation is addition, the second multiplication, third exponentiation, fourth tetration, etc. Is there a meaningful way to define a $1.5^{th}$ operation, something between addition and multiplication. How about a $-1^{th}$ operation. Thanks!

• I think one could equivalently ask: can we extend the Ackermann function to negative- or rational-valued inputs? – Patrick Stevens Dec 31 '16 at 20:36
• Yes. I wonder how that would relate to the extended operations. – mtheorylord Dec 31 '16 at 20:40
• mathoverflow.net/questions/146786/… – Patrick Stevens Dec 31 '16 at 20:41
• These are known as the hyperoperations, and are defined as \begin{align}f_k(x,y)=\begin{cases}xy;&k=1\\x;&y=1\\f_{k-1}(f_k(x,y-1),y)&k>1\text{ and }y>1\end{cases}\end{align} I think this grows faster than Ackermann if you consider it as a three-argument function $f(k, x, y)$. – Challenger5 Dec 31 '16 at 20:44
• $x,y\mapsto \frac{xy+x+y}{2}$ would be "between" addition and multiplication in at least one sense. Can you think of a property you want your 1.5th operation to have but this one doesn't? – Henning Makholm Dec 31 '16 at 20:45