Ember Array Notation

Ember Array Notation Accessed 2023-12-2. is an array notation made by the wikia user Sebastian Gomez05.
Hach stand alone array is written as e(@). e stands for Ember 1056, @ stands for the array placed. However, e(@) is only to express that the array is EAN. Arrays placed inside an array are shown just as (@). Also, e(@,1) = e(@). 
Unfortunately, the creator abandoned this notation. Accessed 2024-8-26. The creator notes "I was too immature and too excited to have a notation on the Googology wiki, even though this notation helpped me learn, I really have no use for this notation due to it being so slow." This notation is a work in progress (due to the work in progress, this notation may be ill-defined for the time being). The creator allows any branch of this notation to be made, it just needs to be declared that it is a branch of the notation. If $ declares a branch of Ember Array Notation, a valid array in that branch would be e(@)$. The original branch can be noted by ON after the array
==Definition==
Informal rule: If a number is before e declaration, multiply the numbers additions between 2 ember declared arrays (e(@)) is done with a plus sign.
So far, this array notation only has 4 parts, with 3 parts included.
*Linear Array Notation (LAN) Limit \(f_{\omega^{\omega}}(n)\)
*Extended Array Notation (ExAN) Limit \(f_{\omega^{\omega^{\omega}}}(n)\)
*Extend Plus Array Notation (EPAN) Limit \(f_{\omega^{\omega^{\omega^{\omega}}}}(n)\)
*Nested Array Notation (NAN) is not yet included. Limit \(f_{\varepsilon_{0}}(n)\)
===Linear Array Notation===
Linear Array Notation is the first, easiest, and weakest part of Ember Array Notation. In this variant @ can only be natural numbers, separated by a comma as an array.
==== Rules ====
* Exponent rule: e(a,b) = a^b
* Collapsing rule: e(a,b,c...). If a=1, total=1. If n=0, total=1. If a=2, total=4. If b=1, total=a (The first entry is a, the second is b, any entry is n)
*2 and 3 entry arrays: 2 entry arrays are e(a,b) = a↑b. 3 entry arrays work like this BAN equivalent. e(a,b,c) = \(\{a, b, c\}\) = a{c}b
*2nd entry recurrsion: The second entry increases by itterating the "last entry" of the array with one argument less. For arrays of 4+. For example: e(a,b,c,d,e,f) = e(a,a,a,a,(a,b-1,c,d,e,f)). e(a,b,c,d) = e(a,a,(a,b-1,c,d)).
*Entry recurrsion: e(a,b,c...). If the previous entries from entry n are the same number, you can rewrite it by keeping a's value, setting b to 2, and setting all other entries to the left of n to 1. n is this entry
==== Examples ====
\(e(3,3,3)\) = \(e(3,(3,2,3),2)\)
\(e(3,3,4)\) = \(e(3,(3,2,4),3)\)
\(e(4,7,6)\) = \(e(4,(4,6,6),5)\)
\(e(3,3,1,2)\) = \(e(3,3,(3,3,3))\)
\(e(5,5,5,5)\) = \(e(5,2,1,1,2)\)
\(e(6,6,6,6,6)\) = \(e(6,2,1,1,1,2)\)
==== Comparison ====
For any entry linear arrays = \(e(a,b,c,d,e,f...)\) = \(\{a, b, c, d, e, f...\}\) using BEAF.
=== Extended Array Notation ===
Extended Array Notation is the second part of Ember Array Notation. It adds higher level separators between the entries. Comma is short for \([1]\). A valid nth order entry (separated by [n]) not only can be a value, but an array with separators as high as the [n-1]. In this variant, @ can be arrays up to the (n-1)th level separated by the nth level separator
==== Rules ====
* Separator tail rule: e(@A1) = e(@) (A is the separator, )
*Collapsing rule: e(a,b,c...). If a=1, total=1. If b=0, total=1. If a=2, total=4. If b=1, total=a
*2nd entry recurrsion: The second entry increases by itterating the "last entry" of the array with one argument less. For arrays of 4+. For example: e(a,b,c,d,e,f[n]o) = e(a,a,a,a,(a,b-1,c,d,e,f[n]o)[n]o). e(a,b,c,d[n]o) = e(a,a,(a,b-1,c,d[n]o)[n]o).
*Entry recurrsion: e(a,b,c...). If the previous entries within a 2nd order entry or first order array from entry n are the same number, you can rewrite it by keeping a's value, setting b to 2, and setting all other entries to the left of n to 1. n is this entry.
*In e(a,a,a,a...[2]b), if b gets to the point of repeating the array from a before the separator, write it as e(a,c[2]1,2) which equals e(a,a,a,a...[2](a,a,a,a...[2]...)) (c of those same level arrays nested c number of times). Also, e(a,a,a,a...[2]1,2) (b times) = e(a,b[2]2,2). Nest them again with this suffix to end up with e(a,b[2]1,3) = e(a,a,a...[2](a,a,a...[2](...),2),2) (b times)
*e(a,a,a,a,a,a,a...[2]n) (b times) can be rewritten as e(a,b[2]n+1). If separated by a higher separator, e(a,a,a,a,a...[n]o) (b times), can be rewritten as e(a,b[2]2[n]o).
*e(a,b[n]2) means that a was written b times per first order array, b first order arrays in second order array and so on for (n-1)th order array. This means the entry a is written b^(n-1) times.
*If a was written b times per order array of n-1 dimentions, but a number was after [n] separator, add 1 to that number.
==== Examples ====
\(e(3,3,3)\) = \(e(3,3[2]2)\)
\(e(3,3,3,3,3,3)\) = \(e(3,6[2]2)\)
\(e(3,65,1,2[2]1)\) = \(e(3,65,1,2)\)
\(e(5,5,5,5,5,5[2]2)\) = \(e(5,6[2]3)\)
\(e(5,5,5,5[2]3)\) = \(e(5,4[2]4)\)
\(...\)
\(e(3,3,3,3,3[2](3,3,3,3,3[2](3,3,3,3,3[2](3,3,3,3,3[2]3))))\) = \(e(3,5[2]1,2)\)
\(e(8,8,8,8,8[2]1,2)\) = \(e(8,5[2]2,2)\)
\(e(4,3[2]1,3)\) = \(e(4,4,4[2](4,4,4[2]4,2),2)\)
\(e(4,3[2]3,3)\) = \(e(4,3[2]3,2,2)\)
\(e(7,4[2]5,7,4)\) = \(e(7,4[2]5,(5,6,4),3)\)
\(e(4,3[2]3,3,3)\) = \(e(4,3[2]3,2,2)\) = \(e(4,3[2]3,2,1,2)\)
\(e(9,8[2]7,4,5,6)\) = \(e(9,8[2]7,7,(7,3,5,6))\)
\(e(9,8[2]7,7,7,3)\) = \(e(9,8[2]7,2,1,4)\)
\(e(8,8,8,8,8,8,8,8,8,8[17]5)\) = \(e(8,10[2]2[17]5)\) (I will show you how to increase n in [n] later on)
\(...\)
\(e(3,3,3,3[2]3,3,3,3[2]3,3,3,3[2]3,3,3,3)\) = \(e(3,4[3]2)\) 
\(e(3,3,3,3[2]3,3,3,3[2]3,3,3,3[2]3,3,3,3[3]5)\) = \(e(3,4[3]6)\)
\(e(7,7[2]7,7[3]7,7[2]7,7)\) = \(e(7,2[4]2)\) 
\(e(8,8[2]8,8[3]8,8[2]8,8[4]8,8[2]8,8[3]8,8[2]8,8)\) = \(e(8,2[5]2)\) 
\(e(8,8[2]8,8[3]8,8[2]8,8[4]8,8[2]8,8[3]8,8[2]8,8[5]3)\) = \(e(8,2[5]4)\)
=== Extend Plus Array Notation ===
Expansive Array Notation is the third part of Ember Array Notation. This time, arrays are allowed to be put inside as separator levels. However, getting the second sentry is not the same as getting it in the normal array. Arrays can be put in the form \(e(a,b[@]c)\) (a,b, and c are constants while @ is a linear array). However, this can only be obtained when the normal entry level gets so powerful that it recurses the array.
* Separator tail rule: e(@A1) = e(@) (A is the separator, )
*Collapsing rule: e(a,b,c...). If a=1, total=1. If b=0, total=1. If a=2, total=4. If b=1, total=a
*2nd entry recurrsion: The second entry increases by itterating the "last entry" of the array with one argument less. For arrays of 4+. For example: e(a,b,c,d,e,f[n,o]x) = e(a,a,a,a,(a,b-1,c,d,e,f[n,o]x)[n,o]x). e(a,b,c,d[n,o]x) = e(a,a,(a,b-1,c,d[n,o]x)[n,o]x).
*Entry recurrsion: e(a,b,c...). If the previous entries within a 2nd order entry or first order array from entry n are the same number, you can rewrite it by keeping a's value, setting b to 2, and setting all other entries to the left of n to 1. n is this entry.
*Entry Array Repetition: If the separator recursses itself in the method of a being the entry used in the entire array and it recursing b number of times, write it like e(a,b[1,2]2), which equals e(a,a[(a,a[(a,a[...]a)]a)]a)]a) b times from inside to out. If it is an entry in any level separator, start counting inward from that separator entry and not the entire array
*Suffix increment: on the previous entry from c in the separator array with all other entries before it to be a (also have the array outside the separator be a's). Repeat this b times. Write it with a and b before the separator, and increase the next entry (c) while reverting the end number to a 2 and the separator entries before c to 1
*First Separator Entry Increment: To increase the first separator entry. write each a entry b times per b orders to get from e(#[c,2]#[c,2]#) (# is b a's per each b levels) to e(a,b[c+1,2]2)
*e(a,a,a,a,a,a,a...[2]n) (b times) can be rewritten as e(a,b[2]n+1). If separated by a higher separator or an array, e(a,a,a,a,a...[n,o]x) (b times), can be rewritten as e(a,b[2]2[n,o]x).
==== Examples ====
\(e(4,4[(4,4[(4,4[(4,4[(4,4[4]4)]4)]4)]4)]4)\) = \(e(4,6[1,2]2)\)
\(e(3,3[(3,3[(3,3[(3,3[(3,3[3]3)]3)]3)]3)]3[1,2]2)\) = \(e(3,6[1,2]3)\)
\(e(5,5[1,2]2)\) = \(e(5,2,2[1,2]2)\)
\(e(5,3,2[1,2]2)\) = \(e(5,5[(5,5[(...)]5)]5)\) (\(e(5,5[(5,5[(5,5[(5,5[5]5)]5)]5)]5)\) times)
\(e(14,15,7[1,2]2)\) = \(e(14,(14,14,7[1,2]2),6[1,2]2)\)
\(e(8,8,8,8,8,8,8,8[1,2]2)\) = \(e(8,8[2]2[1,2]2)\) = \(e(8,2[2]3[1,2]2)\)
\(e(2,2,2[2]2,2,2[2]2,2,2[1,2]5)\) = \(e(2,3[2,2]5)\)
\(e(4,4[(4,4[(4,4[(4,4[(4,4[(4,4[4]4)]4)]4)]4)]4),2]4)\) = \(e(4,7[1,3]2)\)
\(e(4,4[4,(4,4[(4,4[(4,4[(4,4[(4,4[4]4)]4)]4)]4)]4)]4)\) = \(e(4,7[1,1,2]2)\)
\(e(3,3[3,(3,3[(3,3[(3,3[(3,3[(3,3[3]3)]3)]3)]3)]3),2]3)\) = \(e(3,7[1,1,3]2)\)
\(e(4,4[4,4,4,(4,4[(4,4[(4,4[(4,4[4]4)]4)]4)]4)]4)\) = \(e(4,6[1,1,1,2]2)\)
==Sources ==
Category:Functions
Category:Notations
Category:Array Notations
Category:Incomplete