GFR Wrote:GML - The Grand-mother Law, considered by all people approaching this matter for the first time, as a first definition of the hierarchy, after applying the "priority to the right" traffic rules to the hyperops operators. It sounds like:

a[s]<r>a = a[s+1](r+1).

Unfortunately the naming is against its context!

What you call grand mother law is rather the daughter law of the mother law, in the sense that it is a particular case of the mother law for those operations with the initial condition a[s+1]1=a:

From GML directly follows:

a=a[s]<0>a=a[s+1]1

Actually you cant use it to define a zeration because for the above equation is wrong/not applicable. Yes, is wrong. The context here is what you call an equality. You have an equality (your GML) that means that the equation is true for all assignments of the contained variables from their corresponding domain of definition. Then we put the particular value and into the GML and gain the *equality* , i.e. it must be true for all of the domain of definition (which I assume is the real numbers). But instead it is not even true for one ! So the GML is not true for .

Quote:DL - The Daughter Law (this is a ... new one !! Haha! Well, not really!). In fact, the recursive application of the hyperops operator gives, as a consequence of situation "b":

a[s]x = x = a[s+1]oo.

The daugher law should correctly be stated as:

If a[s+1]oo=x then a[s]x=x.

As I already showed in the Zeration thread what you call DL is a consequence of your GML and

the more general DL that follows from the mother law (i.e. is valid for *all* operations not only those with x[s+1]1=x) is

If a[s+1]oo=x then a[s]x=x[s+1]1.