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In this article we investigate whether the addition-free theta functions forma canonical notation system for the linear versions of Friedman'swell-partial-orders with the so-called gap-condition over a finite set oflabels. Rather surprisingly, we can show this is the case for two labels, butnot for more than two labels. To this end, we determine the order type of thenotation systems for addition-free theta functions in terms of ordinals lessthan $\varepsilon_0$. We further show that the maximal order type of theFriedman ordering can be obtained by a certain ordinal notation system which isbased on specific binary theta functions.

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