Examples of nonautostable systems.

*(English. Russian original)*Zbl 0595.03031
Sib. Math. J. 25, 538-545 (1984); translation from Sib. Mat. Zh. 25, No. 4(146), 37-45 (1984).

An enumeration \(\nu\) : \(N\to {\mathfrak A}\) is called a constructivization of a countable model \({\mathfrak A}\) if (roughly speaking) the relations and functions of the presentation of \({\mathfrak A}\) on N given by \(\nu\) are recursive. Constructivizations \(\nu\) and \(\mu\) are autoequivalent if there is an automorphism \(\phi\) of \({\mathfrak A}\) and a recursive function f such that \(\phi \nu =\mu f\). Model \({\mathfrak A}\) is called autostable if any two constructivizations of \({\mathfrak A}\) are autoequivalent. The algorithmic dimension of a model \({\mathfrak A}\), \(\dim_ A{\mathfrak A}\), is the maximal number of nonautoequivalent constructivizations of \({\mathfrak A}\). Among the results on algorithmic dimensions and possible numbers of autoequivalent constructivizations, which are not recursively equivalent, we have the following: there are models \({\mathfrak A}\) and \({\mathfrak B}\) such that \(\dim_ A{\mathfrak A}=1\) and \(\dim_ A{\mathfrak A}^ 2\geq 2\), \(\dim_ A{\mathfrak B}=\aleph_ 0\) and \(\dim_ A{\mathfrak B}^ 2=1\). Examples are given to show that the class of autostable models is not closed under finitely generated extensions. The paper is concluded with remarks concerning autostability of periodic groups.

Reviewer: R.Kossak

##### MSC:

03C57 | Computable structure theory, computable model theory |

03D45 | Theory of numerations, effectively presented structures |

##### Keywords:

autoequivalence; constructivization; algorithmic dimension of a model; autostable models; autostability of periodic groups
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\textit{S. S. Goncharov} and \textit{A. A. Novikov}, Sib. Math. J. 25, 538--545 (1984; Zbl 0595.03031); translation from Sib. Mat. Zh. 25, No. 4(146), 37--45 (1984)

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