Class of algebras of primitive recursive functions

In this paper Robinson's algebra is embedded in a countable class of algebras of primitive recursive functions. Each algebra of this class contains the operations of addition and composition of functions and also one of the operations $i_a$ which are defined as follows: $g(x)=i_af(x)$ ($a=0,1,2,\dots$) if $g(x)$ satisfies the equations $g(0)=a$, $g(x+1)=f(g(x))$. In this paper we study the properties possessed by all or almost all the algebras of this class.