## FANDOM

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I this blog post I'll present some new arrow notations.

## Weak arrow notation

This is based on down arrow notation, but this is really weak.

### Rules

1. $$a \uparrow_{weak} b = a^b$$

2. $$a \uparrow_{weak}^c 1 = a$$

3. $$a \uparrow_{weak}^c b = (a \uparrow_{weak}^c b-1) \uparrow_{weak}^{c-1} b$$

### Values and bounds

$$a \uparrow_{weak}^2 b = a^{b!}$$

$$a \uparrow_{weak}^{c+1} b = a^{b!c}$$

$$a \uparrow_{weak}^{b+1} b = a^{T(b)}$$

$$a \uparrow_{weak}^{a+1} a = a^{T(a)} < a \uparrow\uparrow 4$$

## Mixed Arrow Notation 2.0

The normal rules of chained arrow notation remains unchanged.

a1. $$X \rightarrow\uparrow^{c} a = X \rightarrow\uparrow^{c-1} a \rightarrow\uparrow^{c} a-1$$

a2. $$X \diamond\uparrow Y = X \diamond Y$$

a3. $$X \diamond 1 = X$$

Of course we can extend.

a4. $$X \diamond a \diamond\rightarrow\uparrow^{c} b = X \diamond a \diamond\rightarrow\rightarrow b \diamond\rightarrow\rightarrow c$$

### Analysis

$$a \rightarrow\uparrow b \approx f_{\omega \cdot b}(a)$$

$$a \rightarrow\uparrow b \rightarrow 2 \approx f_{\omega^2}^b(a)$$

$$a \rightarrow\uparrow b \rightarrow c+2 \approx f_{\omega^2+c}^b(a)$$

$$a \rightarrow\uparrow a \rightarrow a \rightarrow 2 \approx f_{\omega^2+\omega+1}(a)$$

$$a \rightarrow\uparrow a \rightarrow\uparrow a \approx f_{\omega^22}(a)$$

$$a \rightarrow\uparrow^{2} b \approx f_{\omega^2 \cdot b}(a)$$

$$a \rightarrow\uparrow^{c} b \approx f_{\omega^c \cdot b}(a)$$