Ordinal notation systems corresponding to Friedman's linearized well-partial-orders with gap-condition

In this article we investigate whether the addition-free theta functions form a canonical notation system for the linear versions of Friedman's well-partial-orders with the so-called gap-condition over a finite set of labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less than $\varepsilon_0$. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions.
Comment: 41 pages