Bilinear spaces over a fixed field are simple unstable

Mark, Kamsma,

doi:10.1016/j.apal.2023.103268

Annals of Pure and Applied Logic

2023

DOI: 10.1016/j.apal.2023.103268

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Bilinear spaces over a fixed field are simple unstable

Kamsma, Mark

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Cited by 3 publications

References 13 publications

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Positive indiscernibles

Kamsma

2024

Arch. Math. Logic

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We generalise various theorems for finding indiscernible trees and arrays to positive logic: based on an existing modelling theorem for s-trees, we prove modelling theorems for str-trees, str$$_0$$ 0 -trees (the reduct of str-trees that forgets the length comparison relation) and arrays. In doing so, we prove stronger versions for basing—rather than locally basing or EM-basing—str-trees on s-trees and str$$_0$$ 0 -trees on str-trees. As an application we show that a thick positive theory has k-$$\mathsf {TP_2}$$ TP 2 iff it has 2-$$\mathsf {TP_2}$$ TP 2

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Positive indiscernibles

Kamsma

2024

Arch. Math. Logic

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Ultrahomogeneous tensor spaces

Harman,

Snowden

2024

Advances in Mathematics

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Dividing Lines Between Positive Theories

DMITRIEVA,

GALLINARO,

KAMSMA

2023

J. symb. log.

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We generalise the properties $\mathsf {OP}$ , $\mathsf {IP}$ , k- $\mathsf {TP}$ , $\mathsf {TP}_{1}$ , k- $\mathsf {TP}_{2}$ , $\mathsf {SOP}_{1}$ , $\mathsf {SOP}_{2}$ , and $\mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having $\mathsf {TP}$ and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has $\mathsf {OP}$ iff it has $\mathsf {IP}$ or $\mathsf {SOP}_{1}$ and that T has $\mathsf {TP}$ iff it has $\mathsf {SOP}_{1}$ or $\mathsf {TP}_{2}$ , analogous to the well-known results in full first-order logic where $\mathsf {SOP}_{1}$ is replaced by $\mathsf {SOP}$ in the former and by $\mathsf {TP}_{1}$ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.

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Bilinear spaces over a fixed field are simple unstable

Cited by 3 publications

References 13 publications

Positive indiscernibles

Positive indiscernibles

Ultrahomogeneous tensor spaces

Dividing Lines Between Positive Theories

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