Effectively Existentially-Atomic Structures

Montalbán, Antonio

doi:10.1007/978-3-319-50062-1_16

Cited by 4 publications

(6 citation statements)

References 14 publications

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“…This is close to the definition of enumeration degree of a structure S as given by Montalbán [25] in the spirit of Knight [26].…”

Section: Strongly Locally Finite Graphssupporting

confidence: 60%

“…For any open‐ended graph

scriptG

, we have that

{prefixDgSp}_{\approx} false(scriptG false) = false{deg (Y) : tr (G) is c.e. in Y false}

. This is close to the definition of enumeration degree of a structure

scriptS

as given by Montalbán in the spirit of Knight . Definition A structure

scriptS

has enumeration degree

X \subseteq ω

{prefixDgSp}_{≅} false(scriptS false) = false{deg (Y) : X is c.e. in Y false}

.…”

Section: Strongly Locally Finite Graphsmentioning

confidence: 88%

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Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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We study degree spectra of structures with respect to the bi‐embeddability relation. The bi‐embeddability spectrum of a structure is the family of Turing degrees of its bi‐embeddable copies. To facilitate our study we introduce the notions of bi‐embeddable triviality and basis of a spectrum. Using bi‐embeddable triviality we show that several known families of degrees are bi‐embeddability spectra of structures. We then characterize the bi‐embeddability spectra of linear orderings and study bases of bi‐embeddability spectra of strongly locally finite graphs.

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“…This is close to the definition of enumeration degree of a structure S as given by Montalbán [25] in the spirit of Knight [26].…”

Section: Strongly Locally Finite Graphssupporting

confidence: 60%

“…For any open‐ended graph

scriptG

, we have that

{prefixDgSp}_{\approx} false(scriptG false) = false{deg (Y) : tr (G) is c.e. in Y false}

. This is close to the definition of enumeration degree of a structure

scriptS

as given by Montalbán in the spirit of Knight . Definition A structure

scriptS

has enumeration degree

X \subseteq ω

{prefixDgSp}_{≅} false(scriptS false) = false{deg (Y) : X is c.e. in Y false}

.…”

Section: Strongly Locally Finite Graphsmentioning

confidence: 88%

Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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“…Theorem 6.5. (ZFC+PD; Montalbán [51]). Let ϕ be an L 1 sentence with uncountably many countable models.…”

Section: Vaught's Conjecturementioning

confidence: 99%

An Introduction to the Scott Complexity of Countable Structures and a Survey of Recent Results

Harrison-Trainor¹

2021

Bull. symb. log

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Every countable structure has a sentence of the infinitary logic which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.

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“…It implies, in particular, that the Borel ≡ -to-≡invariant functions are linearly ordered when compared on a cone. 9 The original conjecture is not only for Borel functions but for all functions although working not over ZFC but over a system denoted by ZF+AD+DC that doesn't include the full axiom of choice, where AD denotes the axiom of determinacy of infinite games and DC a weaker version of the axiom of choice called dependent choice. In that case, we don't get that all functions are iterates of the Turing jump, but that they are well-ordered under comparability on a cone and that the successor operator is the jump.…”

Section: Martin's Conjecturementioning

confidence: 99%

“…The structure of the -degrees is as chaotic as that of the Turing degrees. Kihara and Montalbán proved that the uniformly ≡ -to-≡invariant functions are almost well-ordered under compa- 9 Recall that a linear ordering is a partial ordering where every two elements are comparable.…”

Section: Recent Connections With the Wadge Hierarchymentioning

confidence: 99%