Does any non-admissible numbering form a PCA? Given a numbering $\varphi_0, \varphi_1, \dotsc$ of the unary partial recursive functions, define a PAS as $\mathbb N$ with application $x \cdot y \simeq \varphi_x(y)$. If the numbering is admissible then this PAS is a PCA.
Is there any non-admissible numbering for which this PAS is a PCA?
Admissible numbering: https://en.wikipedia.org/wiki/Admissible_numbering
PAS = partial applicative structure, a set $X$ together with a binary partial operation $x,y \mapsto x \cdot y$ on $X$, written with left associativity.
PCA = partial combinatory algebra, a PAS such that there is $k,s$ such that (1) $k \cdot x \cdot y = x$, (2) $s \cdot x \cdot y$ is always defined and (3) $s \cdot x \cdot y \cdot z \simeq x \cdot z \cdot (y \cdot z)$.
 A: There is no recursive non-admissible numbering for which this PAS is a PCA.
Suppose a recursive numbering $\varphi_0, \varphi_1, \dotsc$ forms a PCA. We show that this numbering is admissible by showing that the s-m-n-theorem for $m=n=1$ holds for this numbering.*
Since
\begin{equation*}
    e, x, y \mapsto \varphi_e(\langle x, y \rangle)
\end{equation*}
is partial recursive there is a representative $a$ in the PCA, i.e. an $a$ such that
\begin{equation*}
    a \cdot e \cdot x \downarrow
\end{equation*}
and
\begin{equation*}
    a \cdot e \cdot x \cdot y \simeq \varphi_e(\langle x, y \rangle)
\end{equation*}
for all natural numbers $e$, $x$ and $y$. Thus
\begin{equation*}
   s(e,x) = a \cdot e \cdot x
\end{equation*}
is a total recursive function such that
\begin{equation*}
   \varphi_{s(e,x)}(y) \simeq \varphi_e(\langle x, y \rangle).
\end{equation*}

*A recursive numbering is admissible if and only if its s-m-n-theorem for $m=n=1$ holds. This is exercise 5.10b on page 26 in:
Soare, Robert I. Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Perspectives in Mathematical Logic. Springer Berlin Heidelberg, 1999.
