Eillion Notation

The is a function created by Googology Wiki user https://sites.google.com/view/rpillagergoogology/eillionnotation.
==Base definition==
* E(n) = 10^(3*10^(3*10^(...(3*10^n)...))+3) with n 3*10's
* E0(n) = E(n)
* Em(n) = Em-1(Em-1(...(Em-1(n))...)) with n sets of brackets
* En(n) = E[1](n) = order type \(\omega\)
==Array notation==
===2-entry array notation===
We can now set forth three rules, where @ is a string of numbers and separators. All numbers in all arrays using this notation must be nonzero positive integers.
* Rule 1. Tailing rule: E[@,1](n) = E[@](n)
* Rule 2. Recursion rule: E[m,k](n) = E[m-1,k](E[m-1,k](...(E[m-1,k](n))...)) with n nests
* Rule 3. Hyperoperation rule: E[n,k](n) = E[1,k+1](n)
The limit of this notation is order type \(\omega^2\).
===Linear array notation===
Now we'll have to edit rule 2. 
* E[m@](n) = E[m-1@](E[m-1@](...(E[m-1@](n))...)) with n nests
* If rule 1, 2, 3 or the rule above does not apply, we start a thing called process which starts from the first number after the opening bracket.
** Case A. If the number is 1, jump to the next entry. 
** Case B. If the number is greater than 1, decrease the entry by 1 and change the previous entry to the number in the () bracket. Check if rules apply. 
** E.g. E[1,2,3](3) = E[3,1,3](3)
The limit of this notation is order type \(\omega^{\omega}\).
==Sources==
Category:Notations
Category:Functions
Category:Incomplete