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Dependent finitely homogneneous rosy structures

Authors

Alf Onshuus, Pierre Simon
Type: Preprint
Publication Date: 2021-01-01
Citations: 0
DOI: https://doi.org/10.48550/arxiv.2107.02727

Abstract

We study finitely homogeneous dependent rosy structures, adapting results of Cherlin, Harrington, and Lachlan proved for $ω$-stable $ω$-categorical structures. In particular, we prove that such structures have finite þ-rank and are coordinatized by a þ-rank 1 set. We show that they admit a distal, finitely axiomatizable, expansion. These results show that there are, up to inter-definability, at most countably many dependent rosy structures M which are homogeneous in a finite relational language.

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References (6)

ℵ0-Categorical, ℵ0-stable structures

1985-03-01
Gregory Cherlin , Louise Harrington , A. H. Lachlan

On countable stable structures which are homogeneous for a finite relational language

1984-09-01
A. H. Lachlan

Totally categorical structures

1989-01-01
Ehud Hrushovski
A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is … A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.
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Interpreting groups in <i>ω</i>-categorical structures

1991-12-01
Dugald Macpherson
Abstract It is shown lhat no infinite group is interpretable in any structure which is homogeneous in a finite relational language. Related questions are discussed for other ω -categorical structures. Abstract It is shown lhat no infinite group is interpretable in any structure which is homogeneous in a finite relational language. Related questions are discussed for other ω -categorical structures.

Properties and consequences of Thorn-independence

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Alf Onshuus
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of … Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.

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