Extended Array Notation is the third part of my array notation.

Definition

The Extended Array Notation has the following form:

a[c X d X e X ... X n]b where b ≥ 1; c,d,e,...,n > 1, and X is the separators. The separators come in many forms, such as {3}, {1,4,5}, {1,1,3{2}2}.

The number a is the base, and b is the iterator.

The {1} separator stands for comma.

Rules

To solve the expression, we need some rules as follows:

• Base Rule: a[0]b = a+1, a[1]b = a+b, a[2]b = ab, a[3]b = a^b.

• Tailing Rule: a[# X 0]b = a[#]b (X can be any separators).

• Prime Rule: a[%]1 = a.

• Recursion Rule: a[c%]b = a[c-1%](a[c%](b-1)), provided c ≥ 0 and b ≥ 1.

• Comma Rule: a[c{1}d%]b = a[c,d%]b.

Followed by some additional rules by following:

1. Reuse the rules for linear arrays to that row (after separators other than commas).

2. a[1{2}c]b = a[1,1,1,...,1,1,2{2}c-1]a with b 1's.

3. a[1{2}1{2}...{2}1{2}c]b = a[1{2}1{2}...{2}1,1,1,...,1,1,2{2}c-1]a with b string of 1's using comma as a separator.

4. a[1{j}1{j}...{j}1{j}c] = a[1{j}1{j}...{j}1{j-1}1{j-1}...{j-1}2{j}c-1]a with b string of 1's using {j-1} separator.

5. a[1{1,1,1,...,1,1,c,d%}2]b = a[1{1,1,1,...,1,b,c-1,d%}2]a, which is different from a[1,1,1,...,1,1,c,d%]b = a[a,a,a,...,a,b,c-1,d%]a.

6. a[1{1{2}c}2]b = a[1{1,1,1,...,1,1,2{2}c-1}2], and so on. Nested arrays reuse the rules 2 through 5.

Explanation

The {2}'s decompose into strings of 1's. This stronger separator becomes the basis of the notation.

Planar arrays

• The smallest array in the form of a[1{2}2]b decomposes into linear array in the form of a[1,1,1,...,1,1,1,2]a with b string of 1's = a[a,a,a,...,a,a,b]a with b entries. FGH level ω^ω.

• a[2{2}2]b = a[1{2}2]a[1{2}2]...[1{2}2]a with b copies of a. FGH level ω^ω+1.

• a[1,2{2}2]b = a[b{2}2]a. FGH level ω^ω+ω.

• a[2,2{2}2]b = a[1,2{2}2]a[1,2{2}2]...[1,2{2}2]a with b copies of a. FGH level ω^ω+ω+1

• a[1,3{2}2]b = a[b,2{2}2]a. FGH level ω^ω+ω2.

• a[1,4{2}2]b = a[b,3{2}2]a. FGH level ω^ω+ω3.

• a[1,1,2{2}2]b = a[a,b{2}2]a. FGH level ω^ω+ω^2.

• a[1,1,1,2{2}2]b = a[a,a,b{2}2]a. FGH level ω^ω+ω^3.

• a[1{2}3]b = a[1,1,1,...,1,1,2{2}2]a with b string of 1's. FGH level (ω^ω)2.

• a[1,2{2}3]b = a[b{2}3]a. FGH level (ω^ω)2+ω.

• a[1{2}4]b = a[1,1,1,...,1,1,2{2}3]a with b string of 1's. FGH level (ω^ω)3.

• a[1{2}5]b = a[1,1,1,...,1,1,2{2}4]a with b string of 1's. FGH level (ω^ω)4.

Now we reached 2-entry second row arrays!

• a[1{2}1,2]b = a[1{2}b]a. FGH level ω^(ω+1).

• a[2{2}1,2]b = a[1{2}1,2]a[1{2}1,2]...[1{2}1,2]a with b copies of a. FGH level ω^(ω+1)+1.

• a[1,2{2}1,2]b = a[b{2}1,2]a. FGH level ω^(ω+1)+ω.

• a[1{2}2,2]b = a[1,1,1,...,1,1,2{2}1,2]a with b string of 1's. FGH level ω^(ω+1)+ω^ω.

• a[1{2}3,2]b = a[1,1,1,...,1,1,2{2}2,2]a with b string of 2's. FGH level ω^(ω+1)+(ω^ω)2.

• a[1{2}1,3]b = a[1{2}b,2]a. FGH level ω^(ω+1)2.

• a[1{2}1,4]b = a[1{2}b,3]a. FGH level ω^(ω+1)3.

• a[1{2}1,1,2]b = a[1{2}a,b]a. FGH level ω^(ω+2).

• a[1{2}2,1,2]b = a[1,1,1,...,1,1,2{2}1,1,2]a with b string of 1's. FGH level ω^(ω+2)+ω^ω.

• a[1{2}1,2,2]b = a[1{2}b,1,2]a. FGH level ω^(ω+2)+ω^(ω+1).

• a[1{2}1,1,3]b = a[1{2}a,b,2]a. FGH level (ω^(ω+2))2.

• a[1{2}1,1,1,2]b = a[1{2}a,a,b]a. FGH level ω^(ω+3).

• a[1{2}1,1,1,1,2]b = a[1{2}a,a,a,b]a. FGH level ω^(ω+4).

Now we reached 3-row arrays!

• a[1{2}1{2}2]b = a[1{2}1,1,1,...,1,1,2]a with b string of 1's. FGH level ω^ω2.

• a[1{2}2{2}2]b = a[1,1,1,...,1,1,2{2}1{2}2]a with b string of 1's. FGH level ω^ω2+ω^ω.

• a[1{2}1,2{2}2]b = a[1{2}b{2}2]a. FGH level ω^ω2+ω^(ω+1).

• a[1{2}1{2}3]b = a[1{2}1,1,1,...,1,1,2{2}2]a with b string of 1's. FGH level (ω^ω2)2.

• a[1{2}1{2}1,2]b = a[1{2}1{2}b]a. FGH level ω^(ω2+1).

• a[1{2}1{2}1{2}2]b = a[1{2}1{2}1,1,1,...,1,1,2]a with b string of 1's. FGH level ω^ω3.

• a[1{2}1{2}1{2}1{2}2]b = a[1{2}1{2}1{2}1,1,1,...,1,1,2]a with b string of 1's. FGH level ω^ω4.

Dimensional arrays

Now we reached dimensional arrays!

• a[1{3}2]b = a[1{2}1{2}...{2}1{2}2]b with b string of 1's separated by {2}. FGH level ω^ω^2.

• a[2{3}2]b = a[1{3}2]a[1{3}2]...[1{3}2]a with b copies of a. FGH level ω^ω^2+1.

• a[1,2{3}2]b = a[b{3}2]a. FGH level ω^ω^2+ω.

• a[1{2}2{3}2]b = a[1,1,1,...,1,1,2{3}2]a with b string of 1's. FGH level ω^ω^2+ω^ω.

• a[1{3}3]b = a[1{2}1{2}...{2}1{2}2{3}2]a with b string of 1's separated by {2}. FGH level (ω^ω^2)2.

• a[1{3}1,2]b = a[1{3}b]a. FGH level ω^(ω^2+1).

• a[1{3}1{2}2]b = a[1{3}1,1,1,...,1,1,2]a with b string of 1's. FGH level ω^(ω^2+ω)

• a[1{3}1{3}2]b = a[1{3}1{2}1{2}...{2}1{2}2]a with b string of 1's separated by {2}. FGH level ω^((ω^2)2).

• a[1{4}2]b = a[1{3}1{3}...{3}1{3}2]a with b string of 1's separated by {3}. FGH level ω^ω^3.

• a[1{5}2]b = a[1{4}1{4}...{4}1{4}2]a with b string of 1's separated by {4}. FGH level ω^ω^4.

Hyperdimensional arrays

Now we reached hyperdimensional arrays that contains commas inside separators!

• a[1{1,2}2]b = a[1{b}2]a. FGH level ω^ω^ω.

• a[1{2}2{1,2}2]b = a[1,1,1,...,1,1,2{1,2}2]a with b string of 1's. FGH level ω^ω^ω+ω^ω.

• a[1{1,2}3]b = a[1{b}2{1,2}2]a. FGH level (ω^ω^ω)2.

• a[1{2,2}2]b = a[1{1,2}1{1,2}...{1,2}1{1,2}2]a with b string of 1's separated by {1,2}. FGH level ω^ω^(ω+1).

• a[1{3,2}2]b = a[1{2,2}1{2,2}...{2,2}1{2,2}2]a with b string of 1's separated by {2,2}. FGH level ω^ω^(ω+2).

• a[1{1,3}2]b = a[1{b,2}2]a. FGH level ω^ω^ω2.

• a[1{1,4}2]b = a[1{b,3}2]a. FGH level ω^ω^ω3.

• a[1{1,1,2}2]b = a[1{1,b}2]a. FGH level ω^ω^ω^2.

• a[1{1,2,2}2]b = a[1{b,1,2}2]a. FGH level ω^ω^(ω^2+ω).

• a[1{1,1,3}2]b = a[1{1,b,2}2]a. FGH level ω^ω^((ω^2)2).

• a[1{1,1,1,2}2]b = a[1{1,1,b}2]a. FGH level ω^ω^ω^3.

Nested arrays

Now we reached nested arrays that reuse the previous rules mentioned above!

• a[1{1{2}2}2]b = a[1{1,1,1,...,1,1,2}2]a with b string of 1's. FGH level ω^ω^ω^ω.

• a[1{2{2}2}2]b = a[1{1{2}2}1{1{2}2}...{1{2}2}1{1{2}2}2]a with b string of 1's separated by {1{2}2}. FGH level ω^ω^(ω^ω+1).

• a[1{1{2}3}2]b = a[1{1,1,1,...,1,1,2{2}2}2]a with b string of 1's. FGH level ω^ω^((ω^ω)2).

• a[1{1{2}1,2}2]b = a[1{1{2}b}2]a. FGH level ω^ω^ω^(ω+1).

• a[1{1{2}1{2}2}2]b = a[1{1{2}1,1,1,...,1,1,2}2]a with b string of 1's. FGH level ω^ω^ω^ω2.

• a[1{1{3}2}2]b = a[1{1{2}1{2}...{2}1{2}2}2]a with b string of 1's separated by {2}. FGH level ω^ω^ω^ω^2.

• a[1{1{1,2}2}2]b = a[1{1{b}2}2]a. FGH level ω^ω^ω^ω^ω.

• a[1{1{1{2}2}2}2]b = a[1{1{1,1,1,...,1,1,2}2}2]a with b string of 1's. FGH level ω^ω^ω^ω^ω^ω.

• ...

And finally, the limit of the nested arrays is n[1{1{...1{1,2}2...}2}2]n, which is an ε0-level.