Positive rudimentarity of the graphs of the Ackermann's and Grzegorczyk's functions

The graphs of the Ackermann's functions $\lambda xyg_n(x,y)$ [3,4] and the Grzegorczyk's functions $f_n$ [2] are shown to be in the class of the positive rudimentary predicates of J. H. Bennett [1]. The latter class is included in the initial class $\mathscr E_*^0$ A. Grzegorczyk [2], thus our result strengthens that of S. V. Pakhomov [5] about the expressibility of the $f_n$'s graphs in $\mathscr E_*^0$. By a generalization of the method applied, the positive rudimentarity of the graph of the Ackerman's function $\lambda nxyg_n(x,y)$ can be proved.