Ivars Wrote:Yes, "minusation" . What about "+- imagination", then? Nice word, anyway. May be we can get "+-imagination" by equalling multiplication and "addition" to "zeration"?How nice indeed .... "imagination" would be. Do you mean something like a[i]x ??
That would lead to "multiplication" becoming equal to "minusation" and that would only be possible if somehow inside multiplication we had "+-imagination"???
In fact, we should have:
y = a[4]x ..... tetration
y = a[3]x ..... exponentiation
y = a[2]x ..... multiplication
y = a[1]x ..... addition
y = a[0]x ..... zeration (GML or ML versions)
y = a[-1]x .... minusation (!!!!!!!)
And, to this, our imagination would add ... "imagination", as:
y = a[i]x ...... imagination.
Why not! However, unfortunately, we cannot make any paraller reasoning. Rank s=i (with my traditional "s" symbol) has to be defined and its properties ... carefully described. But the idea is nice. It was also rapidly mentioned (by KAR and myself), as a remote possibility, in a paper submitted to the WRI Forum. However, the "imagination" name is yours! For the moment, it is only "fantasiation". But, ... who knows?!
Before that, we have to investigate the possiblity of defining fractional ranks of the hyperops hierarchy, if any, starting from the rational half-ranks. Something like:
y = a[1/2]x .... halfation, between zeration and addition
y = a[3/2]x .... sesquition ("sesqui" means: 1 + 1/2), between addition and mutiplication
y = a[5/2]x .... sestration ("sester" means: 1 + 1 +1/2), between multiplication and exponentiation, and .... so on .... !?!
At the level between addition and multiplication (s=1.5), at the "sesquition level", we have a very well known "mean", the Gauss Mean also called the Arithmetic-Geometric Mean. So, perhaps, the corresponding operation (with its mid-way hierarchical chain) could maybe be found. One of these days !!! What would then be the impact of GML, ML, BML and DL? Interesting!!
By the way, do we have any documented proposal for a fractional or an imaginary extensions of the Ackermann function?
GFR