Some properties of graphs of functions in the Grzegorczyk hierarchy

Let $\Gamma^n$ be the set of all primitive recursive functions whose graphs belong to $\varepsilon^n$ [I]. It is proved that $\Gamma^n$ is the closure of $\varepsilon^n$ relative to identification and permutation of variables, to substitution of constants and to special operations I)–4) on p.p. 105–106. In particular $f_n\in\Gamma^0$ for every $n\geq 3$. Here $f_n$ is a modification of Ackermana's function described in [I] p. 30.