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Side nesting (横ネスト in Japanese) is a method to generate a googological system introduced by a Japanese googologist mrna. It admits a generalisation called shifting definition.

## Feature

One of the most characteristic and confusing feature of side nesting is that even if a notation employs $$+$$ as a $$2$$-ary function symbol, it does not necessarily work as the addition. Namely, even if a valid expression $$a$$ corresponds to a countable ordinal $$\alpha$$ and $$a + a$$ is also a valid expression, $$a + a$$ does not necessarily corresponds to $$\alpha + \alpha$$. The characteristic feature of side nesting on $$+$$ is inherited by its generalisation shifting definition. If a notation based on side nesting does not employ $$+$$, then a separator plays the role analogous to $$+$$ in a notation based on side nesting.

## Examples

According to mrna, side nesting is a method which can be found in many notations, while there are few notations which intensionally focus on side nesting. The following three systems are intended to be based on side nesting:

Although none of them has been fully defined, many Japanese googologists get interested to their intended behaviour, because they are expected to be poweful. For the detail explanation of the strength, see the following subsections:

### SSAN

Main article: SSAN

SSAN is the first notation which employs side nesting introduced by a Japanese googologist mrna. It consists of infinite systems 0-SSAN, 1-SSAN, 2-SSAN, …, and ω-SSAN.[1][2][3]

### S-σ

Main article: S-σ

S-σ is a notation introduced by mrna as a system purely based on side nesting, while SSAN also employs another strategy than side nesting.[4]

### Y function

Main article: Y function

Y function has at least 10 versions, created by a Japanese googologist Yukito[5][6][7] None of 10 versions of Y function has been formalised yet.

## References

1. mrna, 七星の剣数.
2. mrna, 絶対秘神の七星剣数.
3. mrna, 二童子達のバックナンバーズ.
5. The user page of Yukito in the Japanese Googology Wiki.
6. Yukito, Y関数, the Japanese Googology Wiki user blog.

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Kanrokoti: KumaKuma ψ function
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By Jason: Irrational arrow notation · δOCF · δφ · ε function
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Nayuta Ito: N primitive
By p進大好きbot: Large Number Garden Number
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Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Other: Taro's multivariable Ackermann function · TR function · Arai's $$\psi$$ · Sushi Kokuu Hen

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