Geodesic
The theory of Lists and $\Sigma$-definability
The Bulletin of Irkutsk State University. Series Mathematics, Tome 4 (2011) no. 4, pp. 27-38

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We consider two-sorted structures (lists algebras) consisting of a basic set $S$ and a set of lists $I_S$ (lists are ordered collections of elements from $S \cup I_S$) with natural relations and operations such as membership relation, head and tail operations etc. and show that recursively definable functions are $\Sigma$-definable in the lists algebras. The recursion is on the length and depth of a list.
Keywords: theory of lists, $\Sigma$-definability, recursion theorem. 
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     title = {The theory of {Lists} and $\Sigma$-definability},
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A. A. Gavryushkina. The theory of Lists and $\Sigma$-definability. The Bulletin of Irkutsk State University. Series Mathematics, Tome 4 (2011) no. 4, pp. 27-38. http://geodesic.mathdoc.fr/item/IIGUM_2011_4_4_a2/