A decomposition of the Rogers semilattice of a family of d.c.e. sets. (English) Zbl 1185.03071

The authors prove the following:
“Main Theorem. There is a family \(\mathcal F\) of d.c.e. sets, and there are computable numberings \(\mu\) and \(\nu\) of the family \(\mathcal F\) such that \(\mu \not\leq \nu\) and such that for any computable numbering \(\pi\) of \(\mathcal F\), either \(\mu\leq\pi\) or \(\pi\leq\nu\). In addition, we can ensure the following:
1) \(\mathcal F\) is a family of c.e. sets and \(\nu\) is a computable numbering of \(\mathcal F\) as a family of c.e. sets;
2) both \(\mu\) and \(\nu\) can be made Friedberg and thus minimal numberings; and so
3) any computable numbering \(\pi\) of \(\mathcal F\) satisfies \(\pi\equiv\nu\) or \(\mu\leq\pi\).”


03D45 Theory of numerations, effectively presented structures
03D25 Recursively (computably) enumerable sets and degrees
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