Abstract
A variant of m-reducibility is introduced using almost polynomial functions, and the resulting partially ordered set \({{\mathcal{M}}_{\mathbb{P}}}\) of the corresponding degrees of undecidability is analyzed. It is proved that the set \({{\mathcal{M}}_{\mathbb{P}}}\) has at least a countable number of minimal elements but no maximal elements. \({{\mathcal{M}}_{\mathbb{P}}}\) is neither an upper nor a lower semilattice. Each element of \({{\mathcal{M}}_{\mathbb{P}}}\), other than the smallest one, can be included in a continuum antichain. We construct a continuum family of pairwise isomorphic initial segments of \({{\mathcal{M}}_{\mathbb{P}}}\), having a countable width and height and intersecting only by the smallest element of the set.
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ACKNOWLEDGMENTS
I am grateful to S.A. Matveev for helpful discussions and to a reviewer for useful comments.
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Translated by V. Arutyunyan
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Marchenkov, S.S. Reducibility by Means of Almost Polynomial Functions. Russ Math. 66, 62–70 (2022). https://doi.org/10.3103/S1066369X2212009X
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DOI: https://doi.org/10.3103/S1066369X2212009X