## 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 $$\mu\leq\pi$$.”

### MSC:

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

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