User blog:Rgetar/Mixed Arrow Notation Extension

This is based on and inspired by Hyp cos' Mixed Arrow Notation.

Let a ⊕ b is any binary operation, for example, a + b, a · b, a ↑↑ b, a ↓↓ b.

Rules:

1. a | b = a ↓ b = a ↑ b = a ↕ b = ab

2. a ⊕| 1 = a ⊕↓ 1 = a ⊕↑ 1 = a ⊕↕ 1 = a

3a. a ⊕| (b + 1) = a ⊕ a

b. a ⊕↓ (b + 1) = (a ⊕↓ b) ⊕ a

c. a ⊕↑ (b + 1) = a ⊕ (a ⊕↑ b)

d. a ⊕↕ (b + 1) = (a ⊕↕ b) ⊕ (a ⊕↕ b)

Variant for ordinals:

3a. a ⊕| b = a ⊕ a

b. a ⊕↓ b = sup((a ⊕↓ c) ⊕ a), c < b

c. a ⊕↑ b = sup(a ⊕ (a ⊕↑ c)), c < b

d. a ⊕↕ b = sup((a ⊕↕ c) ⊕ (a ⊕↕ d)), c, d < b

(At least for a ⊕ b such as if c ≥ a, d ≥ b then c ⊕ d ≥ a ⊕ b).

Variations of rule 2: other numbers instead of 1. For example,

a ⊕| 0 = a ⊕↓ 0 = a ⊕↑ 0 = a ⊕↕ 0 = a

a ⊕| (-1) = a ⊕↓ (-1) = a ⊕↑ (-1) = a ⊕↕ (-1) = a

Examples
a +↑ b = a · b

a ↕↕ 1 = a

a ↕↕ 2 = aa

a ↕↕ 3 = (aa)a a = aa 1 + a

a ↕↕ 4 = (aa 1 + a )a a 1 + a = aa 1 + a + a 1 + a

a ↕↕ 5 = (aa 1 + a + a 1 + a )a a 1 + a + a 1 + a   = aa 1 + a + a 1 + a + a1 + a + a 1 + a

ω ↕↕ ω = ε0

ω ↕↕↕ ω = ζ0

ω ↕↕↕↕ ω = η0

ω ↕↕↕↕↕ ω = φ(4, 0)

ω ↕↕↕↕↕↕ ω = φ(5, 0)

FGH:

fn(a) = a ⊕↕(n) a,

where a ⊕ b = a + 1, and rule 2 modification with 0 is used.

My family of functions [X]a:

[n, b]a = a ⊕↕(n) b,

where a ⊕ b = a + 1 + b, and rule 2 modification with -1 is used.