According to the creator, a@b is defined as
- a@b = a^(ab)b = a^...(a*b arrows)...^b.
- method: solve ab to find amount of arrows and put ab arrows between a and b
Although it is not clarified, the domain of the single @ is perhaps the set of all pair (a,b) of positive integers a and b. The definition includes informal epplipses, which make the notation ambiguous and ill-defined. Instead of the precise definition, the creator adds the example which makes the intention clearer instead:
- 3@3 = 3^(3*3)3 = 3^(9)3 = 3^^^^^^^^^3
This perhaps means that a^(b)c is intended to be defined for any tuple (a,b,c) of positive integers a, b, and c as a ↑b c, and a@b is intended to be defined for any pair (a,b) of positive integers a and b as a ↑ab b.
The creator continues to extend the notation in the following way:
- a@@b = a@b^(a@b)a@b
The problem is that there is no clarified restriction on the associativity, and hence the expression is too ambiguous to parse in a unique way. We simply copy the original extensions below.
in order to solve a@...@b (where the amount of @s are more than 1) deconstruct it into a similar equation of a@b, except replace a and b with a@...(one less than original)...@b.
equation: x^(xy)y =x^…xy arrows…^y
in the case of a@b, x = a and y = b
in the case of a@@b = x=a@b and y = a@b（also works with a@(2)b)
in the case of a@(n)b, x=a@(n-1)b y= a@(n-1)b where a@(n)b = a@...(n @s)...b and n > 1
For the case of a@(n)b where n >1, repeat the n-1 step until you are left with only one @ between all as and bs. Then, use the first case (a@b) to simplify it to have no @s. Clearer example below
We copy several examples of the intended behaviour originally given by the creator. As we explained, the notation has an issue on the ill-definedness, but those examples help us to grasp what the creator wanted to do.
3@@3 = 3@3^(3@3*3@3)3@3 = 3^(9)3^(3^(9)3*3^(9)3 )^3 = (3^^^^^^^^^3) ^...(3^^^^^^^^^3*3^^^^^^^^^3 arrows)...^(3^^^^^^^^^3)
In general, if n>1,
a@(n)b = a@...(n @'s)...@b = a@(n-1)b^(a@(n-1)b*a@(n-1)b)a@b
Simplification example 2
3@(3)3 = 3@@@3 = 3@@3^(3@@3*3@@3)3@@3 = (3@3^(3@3)3@3)^(3@3^(3@3)3@3*3@3^(3@3)3@3b) (3@3^(3@3)3@3)
- mumuji, Cool notation, google site. Retrieved at (UTC) 0:50 28/04/2021.