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Definitions:

The Definitions in it are different from the (fandom) googology wiki community's standards. The (fandom) googology wiki's standards doesn't admitting "Bashicu Matrix System (4) is an ordinal notation". However, these definitions do admitting this "opinion".

Ordinal Notation

A Ordinal Notation is the notation system. It has these properties:

• Format: For example, the format can be described like this: /(\(\d+(,\d+)*\))*/ (can be other format).
• Main Expression: For example, 0,1,2,3,...,n,...
• Successor: The Judge of the Successor Expression of the notation
• Zero: The Judge of The Zero Expression of the notation
• Expand: How to expand Limit Expression.

Well-Defined

A notation is well defined if they have a program to describe, or the definition text is unambiguous.

This "Well-Defined" definition doesn't include "well-ordered".

Limit

A Notation's limit is The largest ordinal that the notation can describe.

The Limit of BM4(Bashicu matrix system ver. 4) is sup{(0)(1,1), (0)(1,1,1), (0)(1,1,1,1)...} since the BMS is well-ordered.

The Limit of Y(Yukito's 1-Y Sequence) is sup{Y(1,3),Y(1,4),Y(1,5)...}, but there are no proofs that proved it's well-ordered or not well-ordered.

The Limit of Parented Predecessor Sequence 1 is
0,1,0,2,0,4,4,3,0,4,3,0,11,10,0,0,10,0,17,9,3,0,21,19,3,0,0,3,0,28,18,9,3,0,33,30,9,3,0,0,3,0,41,29,
18,9,3,0,47,43,18,9,3,0,0,3,0,56,42,29,18,9,3,0,63,58,29,18,9,3,0,0,3,0,73,57,42,29,18,9,3,0,81,75,42,29,18,
9,3,0,0,3,0,92,74,57,42,29,18,9,3,0,101,94,57,42,29,18,9,3,0,0,3,0,113,93,74,57,42,29,18,9,3,0,123,115,74,57,42,
29,18,9,3,0,0,3,0,136,114,93,74,57,42,29,18,9,3,0,147,138,93,74,57,42,29,18,9,3,0,0,3,0,161,137,114,93,74,57,42,29
,18,9,3,0,173,163,114,93,74,57,42,29,18,9,3......
=ζ_0, since the notation has infinite descending chains after this expression, although the notation's main expression is 0,1,2,3,4,...

Analyses

All of them are not strict.