Primitive sequence system

Primitive sequence system (PrSS) is an ordinal notation system defined by BashicuHyudora. It is also a sequence system with sequences of natural numbers, and an expansion system with the base of standard form being \( \{(0,1,2,...,n) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: If \( B \) is the subsequence of \( S \) such that the first element of \( B \) is the last element of \( S \) strictly smaller than the last element of \( S \), and the last element of \( B \) is the second-to-last element of \( S \), and \( G \) is the subsequence of all elements of \( S \) before \( B \), then \( S[n]=G+B+\underbrace{B+B+...+B+B}_n \)

The order type of PrSS is \( \varepsilon_0 \), and there is a perfect correspondence between PrSS and iterated Cantor normal form: If we define the parent of an element \( x \) of a sequence \( S \) as the last element of \( S \) before \( x \) that is smaller than \( x \) (making the first element of \( B \) also the parent of the last element of \( S \) in the definition of expansion), then a map from \( S \) to \( \varepsilon_0 \) can be defined, which maps each element \( x \) to \( \omega^{\alpha_0+\alpha_1+...+\alpha_n} \), where \( \alpha_0,\alpha_1,...,\alpha_n \) are the ordinals to which this maps the elements of \( S \) whose parent is \( x \), in the order in which they appear in \( S \). The order type of the set of sequences smaller than \( S \) is then the sum of ordinals to which the zeroes in \( S \) are mapped. This correspondence follows by transfinite induction.

PrSS is identical to Pair sequence system with all pairs of the form \( (n,0) \), and to one-row Bashicu matrix system.