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

Kossak, Roman; Kotlarski, Henryk; Schmerl, James H.

doi:10.1016/0168-0072(93)90035-c

Cited by 23 publications

(36 citation statements)

References 5 publications

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“…We remark that by a result of Lascar (see [8] and [10]), if M is (countable and) arithmetically saturated, then every open H < G extends to a maximal one.…”

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 97%

“…ForH maximal open there is essentialy exactly one cut stabilized by H; we state the appropriate fact below. The idea of the cut J(H) was used in several papers, e. g., [8] and [13]. Unfortunately, this idea does not work for models of PA with nonstandard definable elements, indeed, in this case there exist open H < G with J(H) invariant (see [15]).…”

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 98%

“…Anyway, in the recursively saturated case, G also has plenty of maximal subgroups. In fact, it is known from [8] that every subgroup of the form 1 or of the form 2 is maximal. A subgroup of the form 4(a) is maximal iff it is the stabilizer of an element whose type is selective (and it is well known that selective and nonselective types are realized in each recursively saturated model).…”

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 98%

“…The description of all maximal subgroups of Aut(M) is as follows. It is implicit in [8] and explicit in [1].…”

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 98%

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Automorphisms of Models of True Arithmetic: More on Subgroups which Extend to a Maximal One Uniquely

Kotlarski¹,

Piekart²

2000

MLQ - Math. Log. Quart.

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“…We remark that by a result of Lascar (see [8] and [10]), if M is (countable and) arithmetically saturated, then every open H < G extends to a maximal one.…”

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 97%

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 98%

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 98%

“…The description of all maximal subgroups of Aut(M) is as follows. It is implicit in [8] and explicit in [1].…”

Section: It Is Easy To Check That N < J(h) < M For H Open and H ⊆ G {mentioning

confidence: 98%

See 2 more Smart Citations

Automorphisms of Models of True Arithmetic: More on Subgroups which Extend to a Maximal One Uniquely

Kotlarski¹,

Piekart²

2000

MLQ - Math. Log. Quart.

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“…M[a]. The moving gaps lemma states that only the trivial automorphism of a countable recursively saturated model of PA fixes all gaps setwise (see Section 3 in [8] or Section 5 in [11]). The original proof of the lemma uses a type MG(x, y) with the following two properties.…”

Section: Automorphisms That Do Not Extendmentioning

confidence: 99%

More on extending automorphisms of models of Peano Arithmetic

Kossak

Kotlarski

2008

Fund. Math.

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Abstract. Continuing the earlier research [Fund. Math. 129 (1988) and 149 (1996)] we give some information about extending automorphisms of models of PA to cofinal extensions.In recent work on automorphisms of recursively saturated models of PA two themes stand out. One is the classification of conjugacy classes of single automorphisms, the other is the classification of subgroups of the automorphism group. Many results in both areas depend on information on how automorphisms of a recursively saturated model can be extended from the model to an elementary extension.Let M be a countable model of Peano Arithmetic, PA, and let N be its cofinal extension. If f is an automorphism of M, g is an automorphism of N and f ⊆ g, then for each a ∈ N , there is b ∈ N such that f (M ∩ a) = M ∩ b. This b is, of course, g(a). Here and elsewhere in the paper we identify elements of models of PA with the sets they code. If for each a ∈ N , there is b ∈ N such that f (M ∩ a) = M ∩ b, we say that f sends coded sets to coded sets. Thus, if f has an extension to an automorphism of N , then both f and f −1 send coded sets to coded ones. We showed in [9] that if M and N are countable and recursively saturated, N is an end elementary extension of M, M = inf{(a) n : n < ω} for all a ∈ N , and f is an automorphism of M such that f and f −1 send coded sets to coded ones, then f can be extended to an automorphism of M. In [10] we showed that the restriction on the type of end extension in the above result is essential. This, more or less, settles the problem of extending automorphisms of recursively saturated models to elementary end extensions. In [10] we considered extendability of automorphisms to cofinal extensions. This is a harder problem.2000 Mathematics Subject Classification: 03C62, 03C50.

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