User blog comment:P進大好きbot/Relation between an OCF and an Ordinal Notation/@comment-32994025-20190811194559/@comment-35470197-20190811224040

If you just emply abbreviations, then it does not effect the uniqueness of the expression of the notation itself. If you consider a notation which has actually two distinct terms sharing the corresponding ordinal, then it is not an ordinal notation by the definition of a total ordering.

In order to create an ordinal notation from such a notation equipped with (not total) recursive partial ordering, you need to determine a subset of "terms of standard form", i.e. a recursive subset satisfying the uniqueness of the expression.