Recursive linear orders with recursive successivities

Moses, Michael

doi:10.1016/0168-0072(84)90028-9

Cited by 27 publications

(5 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , a s } for every s ∈ ω. THEOREM 4 [6]. Let A be a computable structure and R be a k-place computable relation on A.…”

Section: Intrinsically Computable Idealsmentioning

confidence: 99%

Computable ideals in I-algebras

Alaev

2010

Algebra Logic

View full text Add to dashboard Cite

Keywords: Boolean algebra with finite number of distinguished ideals, intrinsically computable ideal, relatively intrinsically computable ideal.We give algebraic descriptions of relatively intrinsically computable ideals in I-algebras (Boolean algebras with a finite number of distinguished ideals) and of intrinsically computable ideals for the case of two distinguished ideals in the language of I-algebras.

show abstract

“…. , a s } for every s ∈ ω. THEOREM 4 [6]. Let A be a computable structure and R be a k-place computable relation on A.…”

Section: Intrinsically Computable Idealsmentioning

confidence: 99%

Computable ideals in I-algebras

Alaev

2010

Algebra Logic

View full text Add to dashboard Cite

show abstract

“…Though not part of the language of linear orders, every linear order has an associated immediate successor relation , where holds for if and only if b is the -immediate successor of a . As explained in [19, Section 3], a computable linear order is -decidable if and only if the immediate successor relation is computable. It is straightforward to check that a computable copy of is computably isomorphic to the usual presentation if and only if is computable.…”

Section: Cohesive Powers Of Computable Copies Ofmentioning

confidence: 99%

“…As explained above, it follows from [19, Section 3] that a computable copy of is computably isomorphic to the usual presentation if and only if is -decidable. Thus we show that if is a -decidable copy of and C is cohesive, then .…”

Section: Cohesive Powers Of Computable Copies Ofmentioning

confidence: 99%

On Cohesive Powers of Linear Orders

DIMITROV¹,

Harizanov²,

Morozov³

et al. 2023

J. symb. log.

View full text Add to dashboard Cite

Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $\omega $ , $\zeta $ , and $\eta $ denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of $\omega $ . If $\mathcal {L}$ is a computable copy of $\omega $ that is computably isomorphic to the usual presentation of $\omega $ , then every cohesive power of $\mathcal {L}$ has order-type $\omega + \zeta \eta $ . However, there are computable copies of $\omega $ , necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to $\omega + \zeta \eta $ . For example, we show that there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \eta $ . Our most general result is that if $X \subseteq \mathbb {N} \setminus \{0\}$ is a Boolean combination of $\Sigma _2$ sets, thought of as a set of finite order-types, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ , where $\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ denotes the shuffle of the order-types in X and the order-type $\omega + \zeta \eta + \omega ^*$ . Furthermore, if X is finite and non-empty, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X)$ .

show abstract

“…Effectively, we mentioned the Remmel-Dzgoev characterization of computably categorical linear orderings in terms of their successor relation. Moses [15] showed that a computable linear ordering L is 1-decidable if and only if the successor relation on L is computable. This is one reason why the complexity of the successor relation on computable linear orderings was studied intensively, in particular in the theorem of Downey, Lempp and Wu mentioned above.…”

Section: The Successor Relationmentioning

confidence: 99%