User talk:180.190.60.236/Mixed arrow notation

The table uses base-3 (trinary) form for use of hyper operators, so we even use the exponents

The total sum of it is $ \sum_{i=1}^n 3^i.$

Variable Notes
There are 3 arrangements in it

->exp(a+(2*pi*i*k)/(log(b)+2*pi*i*n)*(ln(b)+2*pi*i*n)))

->exp((log(b)+2*pi*i*n)/a)

->(log(a)+2*pi*i*k)/(log(b)+2*pi*i*n)

We use subscripts if they are more than one variable of the same letter (example: a,a, it changes to a1,a2)

On tetration, it is similar, but we use subscripts to avoid confusion with others and has a similar periodicity

Tetration
There are $$3^2 = 9$$ and $$3^3 = 27$$ arrangements of compositions

Examples
$$10 \uparrow\uparrow 2 = 10^{10} = 10000000000$$

Inverses
See Inverse functions, here

Extensions
See Extensions, here

Special cases

 * 1) $$a [\text{Right associative tetration}] 2 = a [\text{Left associative tetration}] 2 = a [\text{Mixed tetration}] 2 = a^a$$
 * 2) $$a [\text{Right associative tetration}] 1 = a [\text{Left associative tetration}] 1 = a [\text{Mixed tetration}] 1 = a$$
 * 3) $$a [\text{Right associative tetration}] 0 = a [\text{Left associative tetration}] 0 = a [\text{Mixed tetration}] 0 = \log_a{(a)} = {\frac{log(a)}{log(a)}} = 1$$
 * 4) $$1 [\text{Right associative tetration}] b = 1 [\text{Left associative tetration}] b = 1 [\text{Mixed tetration}] b = 1$$

Pentation
There are $$3^4 = 81$$ and $$3^{3^2} = (3^3)^3 = 19683$$ arrangements of compositions, unlike Tetration

We won't take hexation because it is 3^4 is 81, so it would take long

On special cases, Underline is multivalued

Values
Besides 1, 0 and infinity, the pentation has special values

Special cases
A special case regarding is equal to the base raised to height of tetration

Tetration
a↑↑-1 =0

a↓↓-1 =1/0=Error (for somewhat \frac{log(b)}{log(x)} \text{at} x = 1)

a↑↑0=a↓↓0 =a↕↕0 (for $$b < 0$$ assuming b is an integer)=1

a↑↑1=a↓↓1=a↕↕1=a

1↑↑b=1↓↓b=1↕↕b =1*

0↑↑b (for $$2n<2n\pm1$$)=0↓↓b (for $$n>1$$)=0↕↕b =1*

0↑↑b (for $$2n\pm1<2n\pm2$$)=0↓↓b (for $$n<1$$) =0*

Pentation
[Work in progress]

Higher Hyper-operators
If a notation ends in down-arrow (↓), then, it is left-associative

If a notation ends in up-arrow (↑), then, it is right-associative

If a notation ends in up/down arrows (↕), then, it would be a combination of both left and right associativity, ending up in a ↕ b = (a ↕ (b-1)) ↕ (a ↕ (b-1))

Evaluation
See here

Combining operators
Is it possible to combine a lower operator and higher operator, but even same level or higher operator and lower operator, but results are different

Aside from lower, and higher operators, we have the rules for combining

If the left argument is one, then simplify it into one

If the right argument is one, then retain the number

Left variant
(a↑↑b)↑↑c ~ a↑↑↓3

(a↑↑b)↓↓c

(a↑↑b)↕↕c

(a↓↓b)↑↑c

(a↓↓b)↓↓c ~ a↓↓↓3

(a↓↓b)↕↕c

(a↕↕b)↑↑c

(a↕↕b)↓↓c

(a↕↕b)↕↕c ~ a↕↕↓3

Right variant
a↑↑(b↑↑c) ~ a↑↑↑3

a↑↑(b↓↓c)

a↑↑(b↕↕c)

a↓↓(b↑↑c)

a↓↓(b↓↓c) ~ a↓↓↑3

a↓↓(b↕↕c)

a↕↕(b↓↓c)

a↕↕(b↑↑c)

a↕↕(b↕↕c) ~ a↕↕↑3

Both arguments
(a↑↑b)↑↑(c↑↑d) ~ a↑↑↕3

(a↑↑b)↑↑(c↓↓d)

(a↑↑b)↑↑(c↕↕d)

(a↓↓b)↓↓(c↑↑d)

(a↓↓b)↓↓(c↓↓d) ~ a↓↓↕3

(a↓↓b)↓↓(c↕↕d)

(a↕↕b)↕↕(c↓↓d)

(a↕↕b)↕↕(c↑↑d)

(a↕↕b)↕↕(c↕↕d) ~ a↕↕↕3

Left variant
(a↑↑↑b)↑↑c

(a↑↑↑b)↓↓c

(a↑↑↑b)↕↕c

(a↓↓↓b)↑↑c

(a↓↓↓b)↓↓c

(a↓↓↓b)↕↕c

(a↕↕↕b)↑↑c

(a↕↕↕b)↓↓c

(a↕↕↕b)↕↕c

Right variant
a↑↑↑(b↑↑c)

a↑↑↑(b↓↓c)

a↑↑↑(b↕↕c)

a↓↓↓(b↑↑c)

a↓↓↓(b↓↓c)

a↓↓↓(b↕↕c)

a↕↕↕(b↑↑c)

a↕↕↕(b↓↓c)

a↕↕↕(b↕↕c)

Both arguments, with a middle lower
(a↑↑↑b)↑↑(c↑↑d)

(a↑↑↑b)↑↑(c↓↓d)

(a↑↑↑b)↑↑(c↕↕d)

(a↓↓↓b)↓↓(c↑↑d)

(a↓↓↓b)↓↓(c↓↓d)

(a↓↓↓b)↓↓(c↕↕d)

(a↕↕↕b)↕↕(c↑↑d)

(a↕↕↕b)↕↕(c↓↓d)

(a↕↕↕b)↕↕(c↕↕d)

Both arguments, with a middle higher
(a↑↑↑b)↑↑↑(c↑↑d)

(a↑↑↑b)↑↑↑(c↓↓d)

(a↑↑↑b)↑↑↑(c↕↕d)

(a↓↓↓b)↓↓↓(c↑↑d)

(a↓↓↓b)↓↓↓(c↓↓d)

(a↓↓↓b)↓↓↓(c↕↕d)

(a↕↕↕b)↕↕↕(c↑↑d)

(a↕↕↕b)↕↕↕(c↓↓d)

(a↕↕↕b)↕↕↕(c↕↕d)

Left variant
(a↑↑b)↑↑↑c

(a↑↑b)↓↓↓c

(a↑↑b)↕↕↕c

(a↓↓b)↑↑↑c

(a↓↓b)↓↓↓c

(a↓↓b)↕↕↕c

(a↕↕b)↑↑↑c

(a↕↕b)↓↓↓c

(a↕↕b)↕↕↕c

Right variant
a↑↑(b↑↑↑c)

a↑↑(b↓↓↓c)

a↑↑(b↕↕↕c)

a↓↓(b↑↑↑c)

a↓↓(b↓↓↓c)

a↓↓(b↕↕↕c)

a↕↕(b↑↑↑c)

a↕↕(b↓↓↓c)

a↕↕(b↕↕↕c)

Both arguments, with a middle lower
(a↑↑b)↑↑(c↑↑↑d)

(a↑↑b)↑↑(c↓↓↓d)

(a↑↑b)↑↑(c↕↕↕d)

(a↓↓b)↓↓(c↑↑↑d)

(a↓↓b)↓↓(c↓↓↓d)

(a↓↓b)↓↓(c↕↕↕d)

(a↕↕b)↕↕(c↑↑↑d)

(a↕↕b)↕↕(c↓↓↓d)

(a↕↕b)↕↕(c↕↕↕d)

Both arguments, with a middle upper
(a↑↑b)↑↑↑(c↑↑↑d)

(a↑↑b)↑↑↑(c↓↓↓d)

(a↑↑b)↑↑↑(c↕↕↕d)

(a↓↓b)↓↓↓(c↑↑↑d)

(a↓↓b)↓↓↓(c↓↓↓d)

(a↓↓b)↓↓↓(c↕↕↕d)

(a↕↕b)↕↕↕(c↑↑↑d)

(a↕↕b)↕↕↕(c↓↓↓d)

(a↕↕b)↕↕↕(c↕↕↕d)