##
**Automorphisms of models of true arithmetic: Recognizing some basic open subgroups.**
*(English)*
Zbl 0824.03016

The study of the automorphism group of a structure \(M\) (whether viewed as an abstract group, a permutation group acting on \(M\), or even a topological group with sub-basic open sets of the form \(U^ b_ a= \{g\in \operatorname{Aut}(M): g(a)= b\})\) and the amount of information it conveys concerning \(M\) itself has been of considerable recent interest [see, e.g., Automorphisms of first-order structures, edited by D. Macpherson and the second author (1994; Zbl 0797.00010)]. This study has been proved particularly fruitful when \(M\) is a countable recursively saturated model of PA. The paper under review focuses on models of the most distinguished completion of PA, namely true arithmetic (the truth- set of the standard model of PA). The main problem in this area is to distinguish the basic subgroups of \(\operatorname{Aut}(M)\) (those of the form \(U^ a_ a\)), viewed as a topological group, from the open ones by a purely topological-group-theoretic property.

The authors give properties of this sort, which are satisfied by subclasses of the basic subgroups, namely strongly maximal and maximal ones. The problem of finding a property describing precisely the basic subgroups remains unsolved, but Kossak and Schmerl have recently solved this problem for the special case of arithmetically saturated models (models in which the types which are finitely satisfiable and arithmetical in some set in the standard system of the model are realized, or, equivalently, recursively saturated models in which the standard cut \(\mathbb{N}\) is strong).

The authors give properties of this sort, which are satisfied by subclasses of the basic subgroups, namely strongly maximal and maximal ones. The problem of finding a property describing precisely the basic subgroups remains unsolved, but Kossak and Schmerl have recently solved this problem for the special case of arithmetically saturated models (models in which the types which are finitely satisfiable and arithmetical in some set in the standard system of the model are realized, or, equivalently, recursively saturated models in which the standard cut \(\mathbb{N}\) is strong).

Reviewer: A.Enayat (Washington / D.C.)

### Keywords:

models of arithmetic; automorphism group; true arithmetic; basic subgroups; topological group### Citations:

Zbl 0797.00010
PDF
BibTeX
XML
Cite

\textit{H. Kotlarski} and \textit{R. Kaye}, Notre Dame J. Formal Logic 35, No. 1, 1--14 (1994; Zbl 0824.03016)

Full Text:
DOI

### References:

[1] | Gaifman, H., “Models and types of Peano’s arithmetic,” Annals of Mathematical Logic , vol. 9 (1976), pp. 223–306. · Zbl 0332.02058 |

[2] | Kaye, R., Models of Peano arithmetic , Oxford Logic Guides 15, Oxford University Press, Oxford, 1991. · Zbl 0744.03037 |

[3] | Kaye, R., “A Galois correspondence for countable recursively saturated models of Peano arithmetic,” to appear in Automorphisms of first-order structures , edited by R. Kaye and H. D. Macpherson, Oxford University Press, Oxford, forthcoming. · Zbl 0824.03015 |

[4] | Kaye, R., R. Kossak, and H. Kotlarski, “Automorphisms of recursively saturated models of arithmetic,” Annals of Pure and Applied Logic , vol. 55 (1991), pp. 67–99. · Zbl 0748.03023 |

[5] | Kossak, R., and J. H. Schmerl, “The automorphism group of an arithmetically saturated model of Peano arithmetic,” unpublished manuscript. · Zbl 0905.03024 |

[6] | Kossak, R., H. Kotlarski, and J. H. Schmerl, “On maximal subgroups of the automorphism group of a countable recursively saturated model of PA,” to appear in the Annals of Pure and Applied Logic , forthcoming. · Zbl 0796.03043 |

[7] | Kotlarski, H., “On elementary cuts in recursively saturated models of Peano Arithmetic,” Fundamenta Mathematicae , vol. 120 (1984), pp. 205–222. · Zbl 0572.03016 |

[8] | Smoryński, C., “Back and forth inside a recursively saturated model of arithmetic,” pp. 273–278 in Logic Colloquium ’80 , edited by D. van Dalen et al, North Holland, Amsterdam, 1982. · Zbl 0503.03033 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.