Introduction

In the realm of computability theory, the study of general recursive functions and their properties is fundamental. This article delves into the existence of endomorphisms within the set of general recursive functions, where the image is a proper subset. We explore this concept through the lens of Turing machines and provide concrete examples to elucidate the complexities involved.

General Recursive Functions and Endomorphisms

Let (mathcal{G}) denote the set of all general recursive functions (f: mathbb{N} to mathbb{N}). An endomorphism (g: mathcal{G} to mathcal{G}) is a function that satisfies the following properties:

(g) is injective (one-to-one). (g) preserves composition: for any two functions (f_1, f_2 in mathcal{G}), the composition (g(f_1 circ f_2) g(f_1) circ g(f_2)).

The challenge here is to find an endomorphism (g) that is not an isomorphism, meaning that (g) does not have a two-sided inverse. Additionally, (g) does not need to be computable.

Example: Endomorphism via Turing Machines

One such example of an endomorphism involves the use of Turing machines. Any Turing machine (M) can generate a function (f_M: mathbb{N} to mathbb{N}) that takes an encoding (O) and produces an encoding of (O) composed with (M).

Consider the endomorphism (g: mathcal{G} to mathcal{G}) defined by (f mapsto f_M), where (M) computes (f). This function (g) maps each general recursive function to another function that preserves the composition property:

(gf_1 circ gf_2 g(f_1 circ f_2) f_M circ f_M), which holds true since (M) is the same machine computing (f).

This example highlights the intriguing behavior of endomorphisms in general recursive functions, illustrating that the composition property is preserved even when the intermediate function (f_M) is used.

Further Examples and Observations

Another example involves transforming a function (f: mathbb{N} to mathbb{N}) into a new function (g). Given a general recursive function (f), we can define (g(n) 2f(n)). This transformation ensures that:

If (f(n)) is general recursive, then (g(n)) remains general recursive. Injectivity: (g(n_1) g(n_2) implies 2f(n_1) 2f(n_2) implies f(n_1) f(n_2)). Composition preservation: If (f_1) and (f_2) are different general recursive functions, then (2f_1) and (2f_2) are also different.

However, this endomorphism maps only the general recursive functions that take even values into the image, demonstrating that the image is a proper subset of (mathcal{G}).

Conclusion and Open Questions

The exploration of endomorphisms in general recursive functions opens up a diverse array of questions and examples. While the provided endomorphisms demonstrate specific transformations, there is still much to discover about the broader nature of such mappings.

If you have any additional examples or questions related to endomorphisms and general recursive functions, feel free to share them. The rich tapestry of computability theory continues to intrigue and inspire further research.