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Positive numerations of families with one-valued numerations. (English. Russian original) Zbl 0572.03023
Algebra Logic 22, 345-350 (1983); translation from Algebra Logika 22, No. 5, 481-488 (1983).
A result in the theory of numerations of families of recursively enumerable sets is proved. To state the main result, some definitions are necessary.
A numeration $$\nu$$ :$${\mathbb{N}}\to S$$ of a family S of r.e. sets is computable if $$\{$$ (x,n): $$x\in \nu (n)\}$$ is r.e. - equivalently if there is a recursive function g:$${\mathbb{N}}\to {\mathbb{N}}$$ for which $$\nu (n)=W_{g(n)}$$ for all n. A numeration $$\nu$$ is reducible to numeration $$\mu$$, $$\nu\leq \mu$$, if there is a recursive function f:$${\mathbb{N}}\to {\mathbb{N}}$$ for which $$\nu =\mu f$$; $$\nu$$ is equivalent with $$\mu$$ if $$\nu\leq \mu$$ and $$\mu\leq \nu$$. A computable numeration $$\nu$$ is minimal if $$\mu\leq \nu$$ implies that $$\nu\leq \mu$$ for every computable $$\mu$$ ; $$\nu$$ is smallest if $$\nu\leq \mu$$ for every computable numeration $$\mu$$ of the same family S of r.e. sets; $$\nu$$ is positive if $$\{$$ (n,m): $$\nu (n)=\nu (m)\}$$ is r.e.; $$\nu$$ is one-valued if $$n\neq m$$ implies $$\nu$$ (n)$$\neq \nu (m).$$
The main theorem proved now reads: Each family S admitting a one-valued computable numeration has either a smallest one-valued numeration or countably many nonequivalent positive numerations. This result follows from a lemma proved by a priority argument together with a previous result due to the author. The author has a counterexample showing that the main result cannot be improved to read ”nonequivalent one-valued numerations” in place of ”nonequivalent positive numerations”.
Reviewer: P.Clote

##### MSC:
 03D45 Theory of numerations, effectively presented structures 03D25 Recursively (computably) enumerable sets and degrees 03D20 Recursive functions and relations, subrecursive hierarchies
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##### References:
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