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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
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##### References:
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