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Y function is a sort of functions introduced by a Japanese googologist Yukito, and is a function which employs side nesting.[1][2][3]

## Versions

It has at least 10 versions. One version of Y function is expected to roughly reach (0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1) in Bashicu matrix system version 2.3 under the assumption of its termination.

Another version of Y function called 0-Y is unexpectedly broken, and hence the limit was considered to correspond to $$\zeta_0$$ in fast-growing hierarchy, which is not as big as Yukito had first expected. However, it later turned out to correspond to $$2$$ in fast-growing hierarchy.

The limit of another version of Y function called 1-Y is expected to correspond to $$\psi(I)$$ with respect to an unspecified ordinal collapsing function in fast-growing hierarchy with respect to an unspecified system of fundamental sequences.

None of those versions of Y function has been formalised yet.

## Termination

Yukito conjectured that $$Y(\Omega)+Y_{\omega^3}(Y(\Omega))$$ in one version of Y function safely outgrows the limit of Y sequence on 27/02/2020, and hence Y function was supposed to be a quite strong system if it would be completely formalised. However, Yukito noticed that the method applied to S-σ to avoid a certain problem is not applicable to Y function, and hence clarified that he decided to terminate the project to complete the study of Y function on 20/07/2020.

## Sources

1. The user page of Yukito in the Japanese Googology Wiki.

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's $$\psi$$
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 computation programmes · TR function (I0 function)
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Bashicu matrix system 2.3
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 Naruyoko Naruyo: Y sequence computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence · YY sequence · Y function · ω-Y sequence