Computability on linear orderings enriched with predicates

Abstract

Let L be a quasidiscrete linear ordering. We specify some conditions for the existence of a computable presentation for L or for the structure (L, adj), where adj(x, y) is a predicate distinguishing adjacent elements.

On Categoricity of Scattered Linear Orders Of constructive Ranks

“…where the operator Ψ is from Theorem 3.2 and β = α + 2n − 1, if α = 0, and 1) and [y j t ] (α+n−1) . Now we consider the case, when α +n is a limit ordinal, i.e., n = 0.…”

“…To prove the lemma, we use two steps. Firstly, by [1], there are a 0 ′ -computable structure L ′′ = L ′′ ; < L ′′ , S L ′′ and a 0 ′′ -computable embedding function ψ 1 : L ′ → L ′′ with some properties, in particular, such that if x and y are in the same block then ψ 1 (x) and ψ 1 (y) are in the same block. Since Inf ζ L ′ is 0 ′′ -computable, we can find a 0 ′′ -computable set C containing exactly one element from each block of the linear order L ′′ such that if x ∈ C and belongs to a block of type ω, then x is the left limit point.…”

The Simplest Low Linear Order With No Computable Copies

“…The following two results about low linear orders are worth mentioning. The first result is the result of Thurber, Alaev, and Frolov [18]. They proved that every low -quasidiscrete linear order is computably presentable.…”

Computable Linear Orders and Products

“…It is easy to see that we can build a 0 ′′ -computable presentation T 1 of 1 such that the successor relation and the block relation of T 1 are both 0 ′′ -computable. By the result of Alaev, Thurber, and Frolov in [1], 1 has a computable copy. Again by Theorem 1.2, it follows that 1 • L ∼ = • L has a computable copy, contradicting Condition (3).…”

“…To prove "(3) ⇒ (1)", we note first that must be computable. 1 Indeed, let L be an infinite linear order with a least element and a greatest element. Then a computable presentation of • L contains points t 1 , t 2 such that [t 1 , t 2 ] ∼ = .…”