# What would an “algebraic axiomatization of the partial recursive functions” be?

Hartley Rogers in his "Theory of recursive functions and effective computability" (page 55 in the first edition) writes

"What resemblance types are also isomorphism types? A final answer to this question has not been given. The only cases presently known are the type of the empty partial functions, the type of all constant functions, and the type of all universal partial functions.^1

1 A proof that these are the only cases possible could lead to an interesting algebraic axiomatizations of the partial recursive functions."

What did he mean by "algebraic axiomatization of the partial recursive functions"?

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There is a technical definition of algebraic theory: it is a first-order theory with a fixed number (usually finite, sometimes infinite) of finitary operations with all its axioms expressed as universally quantified equations. For example, the theories of groups, rings, and Lie algebras are all algebraic, while the theory of fields is not. – Zhen Lin Jan 7 '12 at 1:21
Thank you for the reply. Am I right to assume that, so far at least, there is no algebraic theory of partial recursive functions? – Peter Percival Jan 25 '12 at 14:53