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I have devised another type of arrow notation similar to Conway's chained arrow notation. I think it's more straightforward than most of my notations/functions, so it'll be easier to follow.

Definition of \(a\leftrightarrow b\)

  • \(a\leftrightarrow 1\) = \(a\uparrow ^{a}a\) = \(a\underset{a}{\underbrace{\uparrow \uparrow . . . \uparrow \uparrow }}a\)
    • \(a,b\leftrightarrow 1\) = \(a\uparrow ^{b}a\) = \(a\underset{b}{\underbrace{\uparrow \uparrow . . . \uparrow \uparrow }}a\)
    • meaning, 
    • \(a,a\leftrightarrow 1\) = \(a\uparrow ^{a}a\) = \(a\leftrightarrow 1\)
    • \(a\leftrightarrow 1\) is also equal to \(A(n)\) using basic Ackermann function
  • ​\(a\leftrightarrow 2\) = Googological Notation - Left-right Arrow Notation - Representation 1
    • \(a,b,c\leftrightarrow 2\) = Googological Notation - Left-right Arrow Notation - Representation 2
    • \(a,a,c\leftrightarrow 2\) = \(a,c\leftrightarrow 2\) = Googological Notation - Left-right Arrow Notation - Representation 3
  • \(a\leftrightarrow 3\) = Googological Notation - Left-right Arrow Notation - Representation 4
  • \(a\leftrightarrow 4\) = Googological Notation - Left-right Arrow Notation - Representation 5
  • \({a,\underset{b}{\underbrace{a,a,a,...a,a}}}\leftrightarrow b\) = \(a\leftrightarrow b\)
  • \({a,\underset{b}{\underbrace{a,a,a,...a,a,c}}}\leftrightarrow b\) = \(a,c\leftrightarrow b\)
  • \({a,\underset{b}{\underbrace{a,a,a,...a,a,c,d}}}\leftrightarrow b\) = \(a,c,d\leftrightarrow b\)
  • \(a\leftrightarrow b\) = \(a\left \{\left \{b-1\right \}\right \}a\) or \(\left \{ a,a,b-1,2\right \}\) in BEAF, or \(a\rightarrow a\rightarrow a\rightarrow b\) in Chained arrow notation.

Part two

This part is very similar, and works very much the same way as Up-arrow notation, except that there are left-right arrows instead of up arrows.

two arrows

  • \(a\leftrightarrow\leftrightarrow 1\) = a
  • \(a\leftrightarrow\leftrightarrow 2\) = \(a\leftrightarrow a\)
  • \(a\leftrightarrow\leftrightarrow 3\) = \(a\leftrightarrow (a\leftrightarrow a)\)
  • \(a\leftrightarrow\leftrightarrow 4\) = \(a\leftrightarrow (a\leftrightarrow (a\leftrightarrow a))\)
  • \(a\leftrightarrow\leftrightarrow b\)  = \(\underset{b}{\underbrace{a\leftrightarrow (a\leftrightarrow (a\leftrightarrow (a\leftrightarrow ...(a\leftrightarrow (a\leftrightarrow a}})))...)))\)

three arrows

  • \(a\leftrightarrow\leftrightarrow\leftrightarrow 1\)  = a
  • \(a\leftrightarrow\leftrightarrow\leftrightarrow 2\)  = \(a\leftrightarrow\leftrightarrow a\)
  • \(a\leftrightarrow\leftrightarrow\leftrightarrow 3\)  = \(a\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow a)\)
  • \(a\leftrightarrow\leftrightarrow\leftrightarrow 4\)  = \(a\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow a))\)
  • \(a\leftrightarrow\leftrightarrow\leftrightarrow b\)  = \(\underset{b}{\underbrace{a\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow ...(a\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow a}})))...)))\)
  • \(a\leftrightarrow\leftrightarrow\leftrightarrow b\)  can also be written as \(a\leftrightarrow^{3} b\)

four arrows

  • \(a\leftrightarrow\leftrightarrow\leftrightarrow\leftrightarrow 1\)  = \(a\leftrightarrow^{4} 1\) = a
  • \(a\leftrightarrow^{4} 2\) = \(a\leftrightarrow\leftrightarrow\leftrightarrow a\)
  • \(a\leftrightarrow^{4} b\) = \(\underset{b}{\underbrace{a\leftrightarrow\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow\leftrightarrow ...(a\leftrightarrow\leftrightarrow\leftrightarrow (a\leftrightarrow\leftrightarrow\leftrightarrow a}})))...)))\)

multiple arrows

  • \(a\leftrightarrow^{c} b\) = \(a\underset{c}{\underbrace{\leftrightarrow\leftrightarrow...\leftrightarrow\leftrightarrow }}b\) = \(\underset{b}{\underbrace{a\leftrightarrow^{c-1} (a\leftrightarrow^{c-1} (a\leftrightarrow^{c-1} (a\leftrightarrow^{c-1} ...(a\leftrightarrow^{c-1} (a\leftrightarrow^{c-1} a}})))...)))\)
  • \(a\leftrightarrow^{c} b\) = \(a\left \{\left \{\left \{c\right \}\right \}\right \}b\) = \(\left \{a,b,c,3\right \}\) in BEAF

Operators in Left-right arrow notation

  • extension = \(a\leftrightarrow b\)
    • ​9 extended to 2 = \(9\leftrightarrow 2\)

Examples of numbers using Left-right arrow notation

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