Diagonalizable @ Notation

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Diagonalizable @ Notation
Growth rate	(Up to a(@)b) Roughly estimated to be slightly weaker than Linear BEAF
Notations	a(#)(#)...(#)(#)b
Author(s)	AlexJN
Year	2024

Diagonalizable @ Notation (D@N) is a notation developed by Wiki user AlexJN. It is mainly designed to be a more powerful version of X-Sequence Hyper-Exponential Notation. Its main feature is using @ (pronounced 'at'), an analogue to X in the X-Sequence Hyper-Exponential notation, to extend the basic rules of the notation.

It has two currently named parts, Basic @ Notation (B@N) and the self-titled Diagonalizing @ Notation (S@N).

Rules

Terminologies/general rules

• \(a(b)\cdot c=\underbrace{a(b)a(b)...(b)a(b)a}_{c}\)
• \(a(b)^c d=a \underbrace{(b)(b)...(b)(b)}_{c}d\)
• \(a(b)\cdot c\backslash (d)e=\underbrace{a(b)a(b)...(b)a(b)a}_{c}(d)e\)
• An inner is an input or inputs inside parentheses and an outer is any other input. For example, in 4(@)(9)(1)5, @, 9, and 1 are inners, and 4 and 5 are outers.
• Any # (pronounced 'hash') in the ruleset represents any string of inners, including none at all, with an exception in rule 2 of the basic @ notation ruleset.
• All outer inputs must be positive integers, all inner inputs must be either positive integers, @, or an @ with a hyper-operator and positive integer or @ input applied.
• All inners must be sorted greatest to least, with @ being larger than all inputs. For example, an expression with inputs 4 and 5 must be expressed as a(5)(4)b, as a(4)(5)b is an invalid expression.
• All expressions are solved right to left.

Basic @ Notation rules

1. \(a(1)b=a\uparrow^ba\)
2. \(a\#(1)b=a\#\cdot b\) (Note that the # in this case must have at least one inner)
3. \(a\#(b)c=a\#(b-1)^ca\)

Diagonalizing @ Notation rules

1. For non-natural number inputs, \(a(\#)b=a(\#[b])a\)
2. \(@[b]=b\)
3. \(a(\#+c)b=a(\#+c-1)^ba,c>0\)
4. \((\#+b)[n]=\#+b[n]\) for limit ordinal b
5. \(\#\cdot(b+1)[n]=\#\cdot b+\#[n]\)
6. \((\#\cdot b)[n]=\#\cdot b[n]\) for limit ordinal b
7. \(\#^{b+1}[n]=\#^b\cdot\#[n]\)
8. \(\#^b[n]=\#^{b[n]}\) for limit ordinal b
9. \((\#\left\{ b \right\}(c+1))[n]=\#\left\{ b-1 \right\}(\#\left\{ b \right\}c)\)
10. \((\#\left\{ b \right\}c)[n]=\#\left\{ b \right\}c[n]\) for limit ordinal c
11. \((\#\left\{ b \right\}c)[n]=\#\left\{ b[n] \right\}c\) for limit ordinal b
12. \((\#(1)b)[n]=\#\left\{ b \right\}\#\)
13. Define the # from B@N as (v)
14. \((\#( v)(1)b)[n]=\#(v)\cdot b\)
15. \((\#( v)(b)c)[n]=\#(v)(b-1)^c\#\)
16. \((\#(1)c)[n]=\#(1)c[n]\) for limit ordinal c
17. \((\#( v)(1)c)[n]=\#(v)(1)c[n]\) for limit ordinal c
18. \((\#( v)(b)c)[n]=\#(v)(b[n])c\) for limit ordinal b
19. \((\#( v)(b)c)[n]=\#(v)(b)c[n]\) for limit ordinal c
20. In an expression where there are multiple limit ordinals, prioritize:
    1. For an expression a(b)c, b
    2. For an expression a{b}c, b
    3. For an expression a(b)c or a{b}c, c
    4. If none of these apply, use the last limit ordinal and go right to left

Example

• 10(3)2
• =10(3)(3)10
• =10(3)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)10
• =10(3)(2)(2)(2)(2)(2)(2)(2)(2)(2)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)10
• =10(3)(2)(2)(2)(2)(2)(2)(2)(2)(2)(1)(1)(1)(1)(1)(1)(1)(1)(1)10(3)(2)(2)(2)(2)(2)(2)(2)(2)(2)(1)(1)(1)(1)(1)(1)(1)(1)(1)...(3)(2)(2)(2)(2)(2)(2)(2)(2)(2)(1)(1)(1)(1)(1)(1)(1)(1)(1)10(3)(2)(2)(2)(2)(2)(2)(2)(2)(2)(1)(1)(1)(1)(1)(1)(1)(1)(1)10 (10 10s)