I decided to fix the ill-definedness of the UNAN since all 3 alternative definitions are all ill-defined.
Unfortunately, DeepLineMadom pointed out that the fourth alternative (my) definition of UNAN proposed by me has some 'subtle problems'.
Basically, I changed the definition to make the notation will only be valid when all entries are natural numbers (1, 2, 3, ...).
Since the description of the parts beyond nested arrays are quite complicated, we only explain parts of the definitions of the first seven systems.
A valid expression in UNAN's basic array notation is of the form a[c]b, where a, b, and c are natural numbers (1, 2, 3, ...). It has the following rules:
Rule 1 (base rule): a[1]b = a^b.
Rule 2 (prime rule): a[c]1 = a.
Rule 3 (recursion rule): a[c]b = a[c - 1](a[c](b - 1)) if b > 1 and c > 1.
If there are two or more distinct rules to apply to a single expression, the lowest-numbered rule which is applicable and whose result is a valid expression will be applied.
In later parts beyond the basic array notation, there will be a minor difference for the rule 2 (prime rule) as follows:
Rule 2 (prime rule): a[#]1 = a for any array #.
The other rules remain unchanged.
Note that according to the list of numbers, arrays always solve from right to left.
A valid expression in UNAN's 2-entry array notation is of the form a[c, d]b, where a, b, c and d are natural numbers (1, 2, 3, ...). The second part adds a fourth rule:
Rule 1 (base rule): a[1, 1]b = a[1]b = a^b.
Rule 2 (prime rule): a[c, d]1 = a.
Rule 3 (recursion rule): a[c, d]b = a[c - 1, d](a[c, d](b - 1)) if b > 1, c > 1 and d > 1.
Rule 4 (hyperoperation rule): a[1, d]b = a[b, d - 1]a.
The number after the array is called the iterator, and the number before it the base.
A valid expression in UNAN's linear array notation is of the form a[#]b, where a and b are natural numbers (1, 2, 3, ...) and # is an array. The third part generalizes to multiple entries:
Rule 1 (base rule): a[1]b = a^b
Rule 2 (tailing rule): a[#, 1]b = a[#]b
Rule 3 (prime rule): a[%]1 = a
Rule 4 (recursion rule): a[c #]b = a[c - 1 #](a[c #](b-1))
If none of them apply, do the following process:
Case A: If it is one, go to the next entry.
Case B: if the entry is greater than 1, decrease it by 1, then change the previous entry to the iterator, and change the iterator to the base. End the process.
A valid expression in UNAN's planar & dimensional array notations is of the form a[c X d X e ... X n]b, where a, b, c, d, e ..., n are natural numbers (1, 2, 3, ...) and X are the separators. The fourth part adds a new separator, {2}. Case B is changed, the rest remains unchanged.
Case B: If it is greater than one, then:
If there is a comma before it, then decrease it by one, change the previous entry to the iterator, and change the iterator to the base.
If there is a {2} before it, decrease it by one, then change 1{2}m to 1, 1, ..., 1, 2{2}m - 1 with the iterator ones, then change the iterator to the base. End the process.
Case B is changed again in part 5:
Case B: If it is greater than one, then:
If there is a comma before it, then decrease it by one, change the previous entry to the iterator, and change the iterator to the base.
If there is {n} before it where n ≥ 2, replace the previous 1{n}m with 1{n - 1}1{n - 1}1{n - 1} ... {n - 1}1{n}m - 1 with the iterator ones, then change the iterator to the base. End the process.
A comma is the same thing as {1}.
In hyper-dimensional and nested arrays, the rules are the same as each other, but in hyper-dimensional arrays, separators cannot contain separators other than commas. Again, all that changes is case B.
Case B: If it is greater than one, then:
If there is a comma before it, then decrease it by one, change the previous entry to the iterator, and change the iterator to the base
If there is {n #} before it where n ≥ 2, replace the previous 1{n #}m with 1{n - 1 #}1{n - 1 #}1{n - 1 #} ... {n - 1 #}1{n #}m - 1 with the iterator ones, then change the iterator to the base. End the process.
If there is {1 #} before it, go into the separator and start the process from there.