Closed Normal Subgroups of the Automorphism Group of a Saturated Model of Peano Arithmetic

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.

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The Automorphism Group of an Arithmetically Saturated Model of Peano Arithmetic

Journal of the London Mathematical Society ◽

10.1112/jlms/52.2.235 ◽

1995 ◽

Vol 52 (2) ◽

pp. 235-244 ◽

Cited By ~ 10

Author(s):

Roman Kossak ◽

James H. Schmerl

Keyword(s):

Automorphism Group ◽

Peano Arithmetic ◽

Saturated Model

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Remarks on weak notions of saturation in models of Peano arithmetic

Journal of Symbolic Logic ◽

10.2307/2273867 ◽

1987 ◽

Vol 52 (1) ◽

pp. 129-148 ◽

Cited By ~ 3

Author(s):

Matt Kaufmann ◽

James H. Schmerl

Keyword(s):

Peano Arithmetic ◽

Saturated Model ◽

Simple Extension ◽

Minimal Models ◽

Recursively Saturated ◽

Countable Models

This paper is a sequel to our earlier paper [2] entitled Saturation and simple extensions of models of Peano arithmetic. Among other things, we will answer some of the questions that were left open there. In §1 we consider the question of whether there are lofty models of PA which have no recursively saturated, simple extensions. We are still unable to answer this question; but we do show in that section that these models are precisely the lofty models which are not recursively saturated and which are κ-like for some regular κ. In §2 we use diagonal methods to produce minimal models of PA in which the standard cut is recursively definable, and other minimal models in which the standard cut is not recursively definable. In §3 we answer two questions from [2] by exhibiting countable models of PA which, in the terminology of this paper, are uniformly ω-lofty but not continuously ω-lofty and others which are continuously ω-lofty but not recursively saturated. We also construct a model (assuming ◇) which is not recursively saturated but every proper, simple cofinal extension of which is ℵ1-saturated. Finally, in §4 we answer another question from [2] by proving that for regular κ ≥ ℵ1; every κ-saturated model of PA has a κ-saturated proper, simple extension which is not κ+-saturated.Our notation and terminology are quite standard. Anything unfamiliar to the reader and not adequately denned here is probably defined in §1 of [2]. All models considered are models of Peano arithmetic.

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Extensions of homomorphisms between localities

Forum of Mathematics Sigma ◽

10.1017/fms.2021.57 ◽

2021 ◽

Vol 9 ◽

Author(s):

Ellen Henke

Keyword(s):

Automorphism Group ◽

General Definition ◽

Fusion System ◽

Normal Subgroups ◽

General Lemma

Abstract We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal {F}$ depends only on $\mathcal {F}$ as long as the object set of the linking system is $\mathrm {Aut}(\mathcal {F})$ -invariant. This was known to be true for linking systems in Oliver’s definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this article. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.

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Saturation of homogeneous resplendent models

Journal of Symbolic Logic ◽

10.2307/2273957 ◽

1986 ◽

Vol 51 (1) ◽

pp. 222-224 ◽

Cited By ~ 3

Author(s):

Julia F. Knight

Keyword(s):

Continuum Hypothesis ◽

Homogeneous Model ◽

Peano Arithmetic ◽

Saturated Model ◽

Relation Symbol ◽

First Order ◽

Recursively Saturated ◽

Recursively Saturated Model ◽

Saturated Structure ◽

The Continuum

The complete diagram of a structure , denoted by Dc(), is the set of all sentences true in the structure (, a)a∈. A structure is said to be resplendent if for every sentence θ involving a new relation symbol R in addition to symbols occurring in Dc(), if θ is consistent with Dc(), then there is a relation P on such that (see[1]).Baldwin asked whether a homogeneous recursively saturated structure is necessarily resplendent. Here it is shown that this need not be the case. It is shown that if is an uncountable homogeneous resplendent model of an unstable theory, then must be saturated. The proof is related to the proof in [5] that an uncountable homogeneous recursively saturated model of first order Peano arithmetic must be saturated. The example for Baldwin's question is an uncountable homogeneous model for a particular unstable theory, such that is recursively saturated and omits some type. (The continuum hypothesis is needed to show the existence of such a model in power ℵ1.)The proof of the main result requires two lemmas.

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On maximal subgroups of the automorphism group of a countable recursively saturated model of PA

Annals of Pure and Applied Logic ◽

10.1016/0168-0072(93)90035-c ◽

1993 ◽

Vol 65 (2) ◽

pp. 125-148 ◽

Cited By ~ 7

Author(s):

Roman Kossak ◽

Henryk Kotlarski ◽

James H. Schmerl

Keyword(s):

Automorphism Group ◽

Maximal Subgroups ◽

Saturated Model ◽

Recursively Saturated ◽

Recursively Saturated Model

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Automorphism groups of models of Peano arithmetic

Journal of Symbolic Logic ◽

10.2178/jsl/1190150283 ◽

2002 ◽

Vol 67 (4) ◽

pp. 1249-1264 ◽

Cited By ~ 3

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.

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Some highly saturated models of Peano arithmetic

Journal of Symbolic Logic ◽

10.2178/jsl/1190150284 ◽

2002 ◽

Vol 67 (4) ◽

pp. 1265-1273

Author(s):

James H. Schmerl

Keyword(s):

Regular Cardinal ◽

Peano Arithmetic ◽

Saturated Model ◽

Elementary Extension ◽

Saturated Models ◽

The One ◽

Recursively Saturated

Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal and is a λ-saturated model of PA such that ∣M∣ > λ, then has an elementary extension of the same cardinality which is also λ-saturated and which, in addition, is rather classless. The construction in [10] produced a model for which cf() = λ+. We asked in Question 5.1 of [10] what other cofinalities could such a model have. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.

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