Side nesting

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.
== Criterion ==
Since side nesting is a sort of a terminology for a description of a strategy to create a googology notation, there is no fixed formalised definition of "whether a notation admits a structure of side nesting or not". However, mrna is trying to show candidates of sufficient conditions, which helps us to understand what notations can be considered as typical side nesting.mrna, 横ネストについて, Japanese Googology Wiki user blog. Although the unformalised definition of "whether a notation admits a structure of side nesting or not" varies as time goes, such a trial will mildly fix the definition. For example, the original idea explained in #Feature refers only to separators such as the \(+\) symbol, and hence the ordinal notation associated to Buchholz's function was not literally regarded as a notation with side nesting. However, the recent trial allows wider structures and syntaxes, and hence the ordinal notation associated to Buchholz's function can be currently regarded as a notation with side nesting. Note that the notion of side nesting makes sense for a notation, but not for a function. Therefore Buchholz's function itself is not one with 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:
* SSAN
* S-σ
* Y function 
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 ===
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.mrna, 七星の剣数.mrna, 絶対秘神の七星剣数.mrna, 二童子達のバックナンバーズ.
=== 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.mrna, Yｶﾞﾁ解析, Google Spreadsheet.
=== Y function ===
Y function has at least 10 versions, created by a Japanese googologist YukitoThe user page of Yukito in the Japanese Googology Wiki.Yukito, Y関数, the Japanese Googology Wiki user blog.Yukito, 2019年11月18日版0-Y関数+1-Y関数, Google Document. None of 10 versions of Y function has been formalised yet.
== References ==
== See also ==
Category:Notations
Category:Googology in Asia
Category:Higher computable level
Category:Side nesting