scholarly journals An ω-categorical structure with amenable automorphism group

2015 ◽  
Vol 61 (4-5) ◽  
pp. 307-314 ◽  
Author(s):  
Aleksander Ivanov
Keyword(s):  
Automorphism Group ◽  
Categorical Structure

ℵ0-categorical structures with arbitrarily fast growth of algebraic closure

2002 ◽  
Vol 67 (3) ◽  
pp. 897-909
Author(s):  
David M. Evans ◽  
M. E. Pantano
Keyword(s):  
Growth Rate ◽  
Automorphism Group ◽  
Finite Subset ◽  
Growth Rates ◽  
Algebraic Closure ◽  
Fast Growth ◽  
Permutation Groups ◽  
Natural Numbers ◽  
Categorical Structure ◽  
Image Position

Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an ℵ0-categorical structure on the set of unordered n-subsets or on the set of n-tuples of elements of . (Recall that by the Ryll-Nardzewski Theorem, if is countable and ℵ0-categorical, the number of the orbits of its automorphism group Aut() on the set of n-tuples from is finite and equals the number of complete n-types consistent with the theory of .) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of ‘folklore’) is that for any sequence (Kn)n∈ℕ of natural numbers there is a countable ℵ0-categorical structure such that the number of orbits of Aut() on the set of n-tuples from is greater than kn for all n.These investigations suggested the study of the growth rate of another sequence. Let be an ℵ0-categorical structure and X be a finite subset of . Let acl(X) be the algebraic closure of X, that is, the union of the finite X-definable subsets of . Equivalently, this is the union of the finite orbits on of Aut()(X), the pointwise stabiliser of X in Aut(). Define

scholarly journals Reconstruction of homogeneous relational structures

2007 ◽  
Vol 72 (3) ◽  
pp. 792-802 ◽  
Author(s):  
Silvia Barbina ◽  
Dugald Macpherson
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Group Structure ◽  
Baire Category ◽  
Abstract Group ◽  
Relational Structures ◽  
Small Index Property ◽  
Categorical Structure ◽  
Small Index

This paper contains a result on the reconstruction of certain homogeneous transitive ω-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly non-standard sense, coming from Baire category).Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut() of an ω-categorical structure determines the topology on Aut(), and hence determines up to bi-interpretability, by [1]; they can also give conditions under which the abstract group Aut() determines the permutation group ⟨Aut (), ⟩. so determines up to bi-definability. One such condition has been identified by M. Rubin in [12], and it is related to the definability, in Aut(), of point stabilisers. If the condition holds, the structure is said to have a weak ∀∃ interpretation, and Aut() determines up to bi-interpretability or, in some cases, up to bi-definability.A better-known approach to reconstruction is via the ‘small index property’: an ω-categorical stucture has the small index property if any subgroup of Aut() of index less than is open. This guarantees that the abstract group structure of Aut() determines the topology, so if is ω-categorical with Aut() ≅ Aut() then and are bi-interpretable.

scholarly journals Coding in the automorphism group of a computably categorical structure

2020 ◽  
Vol 20 (03) ◽  
pp. 2050016 ◽  
Author(s):  
Dan Turetsky
Keyword(s):  
Automorphism Group ◽  
Spectral Dimension ◽  
New Techniques ◽  
Categoricity Spectrum ◽  
Categorical Structure ◽  
Degree Of Categoricity ◽  
Scott Rank ◽  
Computably Categorical Structure

Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.

Jordan permutation groups and limits of 𝐷-relations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asma Ibrahim Almazaydeh ◽  
Dugald Macpherson
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Categorical Structure

Abstract We construct via Fraïssé amalgamation an 𝜔-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a “limit of 𝐷-relations”. The construction is based on a semilinear order whose elements are labelled by sets carrying a 𝐷-relation, with strong coherence conditions governing how these 𝐷-sets are inter-related.

Spontaneous Discovery and Use of Categorical Structure

1993 ◽  
Author(s):  
John P. Clapper ◽  
Gordon H. Bower
Keyword(s):  
Categorical Structure

scholarly journals On the automorphism group of a symplectic half-flat 6-manifold

2019 ◽  
Vol 31 (1) ◽  
pp. 265-273
Author(s):  
Fabio Podestà ◽  
Alberto Raffero
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Group Action ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of {\mathrm{SO}(4)} .

scholarly journals ON EDGE-PRIMITIVE GRAPHS WITH SOLUBLE EDGE-STABILIZERS

2021 ◽  
pp. 1-28
Author(s):  
HUA HAN ◽  
HONG CI LIAO ◽  
ZAI PING LU
Keyword(s):  
Automorphism Group

Abstract A graph is edge-primitive if its automorphism group acts primitively on the edge set, and $2$ -arc-transitive if its automorphism group acts transitively on the set of $2$ -arcs. In this paper, we present a classification for those edge-primitive graphs that are $2$ -arc-transitive and have soluble edge-stabilizers.

scholarly journals Affine cones over cubic surfaces are flexible in codimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Perepechko
Keyword(s):  
Automorphism Group ◽  
Open Subset ◽  
Pezzo Surface ◽  
Ample Divisor ◽  
Transitive Action ◽  
Del Pezzo Surface ◽  
Codimension One ◽  
Cubic Surfaces ◽  
Special Automorphism ◽  
Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

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