On the Structure of Punctual Linear Orders Isomorphic to the Least Limit Ordinal

Abstract

The algorithmic complexity of presentations for various structures receives significant attention in modern literature. The main tool for determining such complexities is reducibility. It is a mapping that preserves relations of signature (for example, equivalence relation, orders, and so on). This work is dedicated to the study of punctual representations of the least limit ordinal with respect to primitive recursive reducibility. We denote this structure as PR(ω). In particular, the paper examines the properties of structures Ω, consisting of computable copies of the least limit ordinal with respect to computable reducibility, and Peq, consisting of punctual equivalence relations with respect to primitive recursive reducibility. We say that the linear order L is reducible to the linear order R, if there exists a total function ρ such that (χ, γ) Є L if and only if (ρ(χ), ρ(γ)) Є R. Reducibility is called computable (primitive recursive) if the function that performs the reducibility is computable (primitive recursive). It is shown that the degree of ω is not the least degree in PR(ω), as it was in Ω. The structure PR(ω) does not contain maximal degrees, and this structure is not dense. Also, an example of an incomparable pair that has the least upper bound is given.