Rigid homogeneous chains

1981 ◽  
Vol 89 (1) ◽  
pp. 07-17 ◽  
Author(s):  
A. M. W. Glass ◽  
Yuri Gurevich ◽  
W. Charles Holland ◽  
Saharon Shelah
Keyword(s):  
Automorphism Group ◽  
General Problem ◽  
Automorphism Groups ◽  
Ordered Sets ◽  
First Order ◽  
Image Position

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S(Ω), satisfiesthen Ω has cardinality at most ℵ0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S(Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o-primitive (i.e. it may have convex congruences). Fortunately, ‘o-primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o-primitive; the second is to determine how the o-primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

Automorphism groups of models of Peano arithmetic

2002 ◽  
Vol 67 (4) ◽  
pp. 1249-1264 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Closed Subgroup ◽  
Automorphism Groups ◽  
Ordered Set ◽  
Peano Arithmetic ◽  
Ordered Sets ◽  
Torsion Free ◽  
Linearly Ordered Set

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.For any structure , let Aut() be its automorphism group. There are groups which are not isomorphic to any model = (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be torsion-free. However, as will be proved in this paper, if (A, <) is a linearly ordered set and G is a subgroup of Aut((A, <)), then there are models of PA such that Aut() ≅ G.If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of A be the basic open subgroups. If ′ is an expansion of , then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroup G ≤ Aut() there is an expansion ′ of such that Aut(′) = G. Thus, if is a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even a closed subgroup of Aut((N, ′)).There is a characterization, due to Cohn [2] and to Conrad [3], of those groups G which are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.

scholarly journals COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS, FREE AMALGAMATION AND AUTOMORPHISM GROUPS

2019 ◽  
Vol 85 (1) ◽  
pp. 199-223 ◽  
Author(s):  
DAOUD SINIORA ◽  
SŁAWOMIR SOLECKI
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Extension Property ◽  
Finite Subgroup ◽  
Homogeneous Structure ◽  
Extension Theorems ◽  
Locally Finite ◽  
Finite Structures ◽  
Image Position ◽  
Fraïssé Class

AbstractWe give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

On a Class of One-Relator Groups

1980 ◽  
Vol 32 (2) ◽  
pp. 414-420 ◽  
Author(s):  
A. M. Brunner
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Image Position ◽  
The University ◽  
Finitely Presented

In this paper, we consider the class of groups G(l, m; k) which are defined by the presentationwhere k, l, m are integers, and |l| > m > 0, k > 0. Groups in this class possess many properties which seem unusual, especially for one-relator groups. The basis for the results obtained below is the determination of endomorphisms.For certain of the groups, we are able to calculate their automorphism groups. One consequence of this is to produce examples of one-relator groups with infinitely generated automorphism groups. This answers a question raised by G. Baumslag (in a colloquium lecture at the University of Waterloo). Our examples are, perhaps, the simplest possible; J. Lewin [10] has found an example of a finitely presented group with an infinitely generated automorphism group.

scholarly journals Permutation groups with small orbit growth

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manuel Bodirsky ◽  
Bertalan Bodor
Keyword(s):  
Automorphism Group ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Permutation Groups ◽  
First Order ◽  
Finite Covers

Abstract Let K exp + \mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most c ⁢ n d ⁢ n cn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c , d c,d with d < 1 d<1 . We show that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from K exp + \mathcal{K}_{{\operatorname{exp}}{+}} . We also show that Thomas’ conjecture holds for K exp + \mathcal{K}_{{\operatorname{exp}}{+}} : all structures in K exp + \mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

1980 ◽  
Vol 45 (2) ◽  
pp. 265-283 ◽  
Author(s):  
Matatyahu Rubin ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Automorphism Groups ◽  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Countably Compact

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

scholarly journals ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 951-971
Author(s):  
NADAV MEIR
Keyword(s):  
New Products ◽  
Elementary Substructure ◽  
First Order ◽  
Order Language ◽  
Image Position

AbstractWe say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

scholarly journals On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains

2016 ◽  
Vol 59 (2) ◽  
pp. 346-353
Author(s):  
Steven Krantz
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Holomorphic Functions ◽  
Biholomorphic Equivalence ◽  
Algebras Of Holomorphic Functions

AbstractWe study and generalize a classical theoremof L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group

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