Analogues of Primitive Recursive Functions Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like to single out a class of $\Pi_{2}$-sentences (with parameters) about $\mathbf{A}$ which reduces to the class of $\Pi_{2}$-theorems of PRA when specialized to the case where $\mathbf{A}$ is the set of hereditarily finite sets. In particular, I would like to single out a special class of "provably total" $\Sigma_1$-definable functions on $\mathbf{A}$, which reduces to the class of primitive recursive functions when $\mathbf{A} =$ hereditarily finite sets.
I would be grateful for any pointers to relevant literature. If it helps, I am primarily interested in the case where $\mathbf{A}$ is the smallest admissible set containing some mathematical structure $M$ (that is, $\mathbf{A} = HYP_M$, in the notation of Barwise's book).
 A: (This is more of a comment than an answer, but it's a bit too long to be split into comments so I'll post it as an answer.)
I don't know about functions defined on an arbitrary admissible set, but at least for admissible levels of the constructible hierarchy, you might be interested in what are called "$(\infty,0)$-recursive functions" in chapter VIII ("Recursion on Ordinals") of Peter G. Hinman's book Recursion-Theoretic Hierarchies (1978, available here), and also on this related question I asked a while ago while trying to make sense (without much success) of the various definitions.  Hinman writes (op.cit., p.378) that:

The $(\infty,0)$-recursive functions will play somewhat the role here of the primitive recursive functions of ordinary recursion theory.

Perhaps even more relevant to your question would be the $(\infty,\lambda)$-recursive functions in Hinman's terminology, or even more the primitive $(\infty,\lambda)$-recursive functions in the terminology of the question I linked to, where $\lambda$ is the height of the admissible set considered (at least for a $L_\lambda$).  But as I noted, the precise relation between these concepts escapes me.
Also somewhat relevant to your question might be Stephen G. Simpson's paper titled "Short Course on Admissible Recursion Theory" on p. 355–390 of Fenstad, Gandy & Sacks (eds.), Generalized Recursion Theory II (1978), proceedings of a symposium held in Oslo in 1977.  It contains the clearest (if terse) explanation I found so far of how primitive recursive ordinal functions are defined and how they relate to more general recursion on ordinals.
A: Actually, for the case $\mathbb A$=$\mathbb H$$Y$$P_{\mathfrak M}$ (which seems, if I am understanding your question correctly, what you are referring to), since $\mathbb H$$F_{\mathfrak M}$$\subseteq$$\mathbb H$$Y$$P_{\mathfrak M}$, the generalization of primitive recursive function you get (rather than want, possibly) are the primitive computable functions of Moschovakis which are defined in his papers "Abstract First-Order Computability, I" and "Abstract First-Order computability,  II".  In fact, Thm. 2.6 in Chapter II of Barwise's book states that the $\Sigma_1$ relations on $\mathbb H$$F_{\mathfrak M}$ are semi-search computable on $\mathfrak M$ and the $\Delta_1$ relations on $\mathbb H$$F_{\mathfrak M}$ are the search-computable relations on $\mathfrak M$ (the search-computable relations being Moschovakis' generalization of the general recursive functions on $\mathfrak N$=(N,+,$\cdot$)) so what you seem to require, that is,  that there exists a special class of "provably total" $\Sigma_1$ functions on $\mathbb A$ which reduces to the class of primitive recursive functions when $\mathbb A$=$\mathbb H$$F_{\mathfrak M}$ does not exist.  That is why I asked those particular questions in my comment to you.  If you believe my observation is incorrect, please be so kind as to show me how the class of functions you are interested in can exist.
