Co-ordering and Type 2 co-ordering

TL;DR

This paper extends the concept of enumeration order reducibility from natural numbers to rational numbers, exploring the classification of c.e. sets on Q and their associated order types, building on prior work on natural numbers.

Contribution

It redefines co-ordering for rational numbers and investigates the variety of c.e. sets on Q, applying existing theories in this new domain.

Findings

Redefinition of co-ordering for rational numbers

Investigation of enumeration order types on Q

Parallel application of theories from natural numbers

Abstract

In [arXiv:1006.4939] the enumeration order reducibility is defined on natural numbers. For a c.e. set A, [A] denoted the class of all subsets of natural numbers which are co-order with A. In definition 5 we redefine co-ordering for rational numbers. One of the main questions there, was: "For a specific c.e. set A, consider set of all enumerations of it which is generated by some Turing machine {TM_A} what are the associated order types in [A]?" Here, we propose the same question for rational numbers, and we try to investigate the varieties of c.e. sets on Q. The theories here are hold for R_c and we could repeat the same theories in this domain, in a parallel way.