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The Extended Arrow Notation is a notation I coined myself to extend Up-Arrow Notation (Arrow Notation). Hope you guys like it.

Basic Definition

This is the basic definition of Extended Arrow Notation.

a^(n)^b = a^^^...^^^b w/ n arrows (^ is an arrow)

Easy, eh?

Examples

10^(100)^10 = 10^^^...^^^10 w/ 100 arrows

or in full expansion:

10^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^10.

41^(11)^88 = 41^^^^^^^^^^^88

etc.

Extensions

Double Parenthesis

We can define an Double Parenthesis operator as a nesting of the normal definition:

a^((n))^b = a^(n^(n^(...(n^(n^(n)^n)^n)...)^n)^n)^b w/ n levels

Triple Parenthesis

We can define triple parenthesis operator:

a^(((n)))^b = a^((n^((n^((n^((..((n^((n^((n))^n))^n))..))^n))^n))^n))^b w/ n levels

where a^((n^((n))^n))^b is a nest of the normal definition

Multiple Parenthesis

We can define the 4 parenthesis operator as a nest of the triple parenthesis operator, the 5 parenthesis operator as a nest of the 4 parenthesis operator, etc.

The limit would be a^(((...(((n)))...)))^b

Exponentiated Parenthesis

We can exponentiate parenthesis:

a^(^(n)^)^b = a^(((...(((n)))...)))^b w/ n parenthesis

We can also define the (^(())^) operator as a nest of the (^()^) operator, the (^((()))^) operator as a nest of the (^(())^) operator, etc.

Multi Exponentiated Parenthesis

We can continue with double exponentiated parenthesis:

a^(^(^(n)^)^)^b = a^(^(((...(((n)))...)))^)^b w/ n parenthesis

We can continue with a^(^(^((n))^)^)^b as a nest of the previous number, and then we can have the (^(^((()))^)^) operator, etc.

We can continue with the triple exponentiating operator as a nest of parenthesis in the double exponentiating level, and then have a 4th level, 5th level, etc.

Tetrated Parenthesis

We can tetrate parenthesis.

a^(^^(n)^^)^b = a^(^(^(^(...(^(^(^(n)^)^)^)...)^)^)^)^b w/ n levels

We can define the (^^(())^^) operator as a nest of the (^^()^^) operator, and then have (^^((()))^^), (^^(((())))^^), etc.

We can also have exponentiated tetrated parenthesis, double exponentiated tetrated parenthesis, multi exponentiated tetrated parenthesis, etc.

Tetrated Exponentiated Tetrated Parenthesis

You did not wait that, did you?

We define:

a^(^^(^(^^(n)^^)^)^^)^b = a^(^^(^(^(^(...(^(^(^(n)^)^)^)...)^)^)^)^^)^b w/ n levels after (^^()^^)

Or something like that.

We can also have the (^^(^(^^(^(^^()^^)^)^^)^)^^) operator (using the other operators we made (for example, we can make a (^^(^(^^(^(^((()))^)^)^^)^)^^) operator)), etc.

Tetrated Tetrated Parenthesis

We define:

a^(^^(^^(n)^^)^^)^b = a^(^^(^(^^(^(^^(^(^^(...(^(^^(^(^^(^(^^(n)^^)^)^^)^)^^)^)...)^^)^)^^)^)^^)^)^^)^b w/ n (^(^^()^^)^) operator levels after (^^()^^)

Or something like that.

We can also recycle our old operators to make operators like (^^(^^(^^(^^(^(^^(^(^^(^(^(^(^(((())))^)^)^)^)^^)^)^^)^)^^)^^)^^)^^) (in 4-level tetrated parenthesis (you get the point)), etc

Multi-arrow Parenthesis

We can extend the notation to pentation defining (^^^(^(^^^()^^^)^)^^^) as a nest of (^(^^()^^)^) levels after (^^^()^^^) and (^^^(^^(^^^()^^^)^^)^^^) as a nest of (^^()^^) after (^^^()^^^). In fact, you get the point. We can extend it to hexation, heptation, octation, n-ation, etc.

Extended Arrow Parenthesis

We can define:

a^(^(m)^(n)^(m)^)^b = a^(^^^...^^^(n)^^^...^^^)^b w/ m arrows.

We can also mix this extension with the other recycled operators (like this, for example: (^(((m)))^(n)^(((m)))^)).

This will give us to nesting of the ^()^ separators (i mean, things like (^(^()^()^()^)^(^()^()^()^)^(^()^()^()^)^))

Double Arrow Extension (Nested Arrow Parenthesis)

We can define:

a^^(n)^^b = n nest of ^()^ operator, being n the value of all ()'s

We can also mix the ^^()^^ operator with our old operators (for example, the ^^(^^()^^(^^^^(^^^(^^(^(^^(^(^^(^(^(^(^(^(((((((((())))))))))^)^)^)^)^)^^)^)^^)^)^^)^^^)^^^^)^^()^^) operator)

Multiple Arrow Extension

We can define the ^^^()^^^ operator as a nest of the ^^()^^ operator in ^^()^^ arrays. Then, we can continue with hexation, heptation, octation, n-ation, etc.

Nested Arrow Extension

We can define the ^()^()^()^ operator as multiple arrows outside the main parenthesis (in the center). Then, we can have ^(())^()^(())^, and then recycle all of our old operators to make more possibilities. Eventually, we will get into things like ^()^()^()^()^()^()^()^, and that will start to be annoying.

The limit is nesting of ^()^.

Beyond Nested Arrow Extension

These extensions have been made by other users and converted to reality.

If you have any extension for my notation, post it in the comment.

Two Parenthesis

This was suggested by googleaarex, and I converted it to reality, but stronger.

First, let define:

a^(n)(2)^b = a^(n)^(n)^(n)^...^(n)^(n)^(n)^b w/ 2^n-1 nestings of ^(n)^

a^((n))(2)^b = a^(n^(n^(...(n^(n^(n)(2)^n)(2)^n)...)(2)^n)(2)^n)(2)^b w/ n levels

etc.

We can reuse our old operators to get bigger numbers.

Next we have:

a^(n)(3)^b = a^(^(n^(^(n^(...(n^(n^(n)(2)^n)(2)^n)...)(2)^n)(2)^n)(2)^(n)^...^)^b w/ n levels (the last ^...^ space was made by me because i don't want to copy that)

etc.

Finally, we have the ^(n)((n))^ operator, etc.

(sorry, too lazy)

Three Parenthesis

We define:

a^(n)(1)(2)^b = a^(n)(^(n)(...(^(n)(2)^)...)(2)^)^b w/ n levels

We can also define the ^(n)(2)(2)^ operator, the ^(n)(1)(3)^ operator, etc.

Multiple Parenthesis

We can also have a four parenthesis operator as a nest of the third parenthesis, etc.

Curly Bracket

Note: This is from me.

We can define:

a^{n}^b = a^(b)(b)(b)...(b)(b)(b)^a w/ n parenthesis

We can have the ^{()}^ operator, etc.

Curly Bracket Operators

We define:

a^^b = 

-- UNDER CONSTRUCTION --

FGH

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