Abstract
We study the structure of 1-degrees inside m-degrees and prove that every \(\Delta_{2}^{0}\) m-degree that has more than one 1-degree contains an infinite antichain of 1-degrees. This strengthens Degtev’s result on computably enumerable m-degrees and gives partial answer to the following question stated by Odifreddi: if an m-degree has more than one 1-degree, does it contain an infinite antichain of 1-degrees? The proved result demonstrates that the answer is positive for \(\Delta_{2}^{0}\) m-degrees.
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REFERENCES
I. I. Batyrshin, ‘‘Q-reducibility and m-reducibility on computably enumerable sets,’’ Sib. Math. J. 55, 995–1008 (2014).
I. I. Batyrshin, ‘‘Irreducible, singular, and contiguous degrees,’’ Algebra Logic 56, 181–196 (2017).
I. Chitaia, R. Omanadze, and A. Sorbi, ‘‘Notes on conjunctive and Quasi degrees,’’ J. Logic Comput. 31, 1317–1329 (2021).
A. N. Degtev, ‘‘\(tt\)- and \(m\)-degrees,’’ Algebra Logic 12, 78–89 (1973).
A. N. Degtev, ‘‘Partially ordered sets of \(1\)-degrees contained in recursively enumerable \(m\)-degrees,’’ Algebra Logic 15, 153–164 (1976).
R. G. Downey, ‘‘On irreducible m-degrees,’’ Rend. Sem. Mat. Univ. Pol. Torino 51, 109–112 (1993).
Y. L. Ershov, ‘‘Positive equivalences,’’ Algebra Logic 12, 620–650 (1971).
J. Myhill, ‘‘Creative sets,’’ Zeit. Math. Log. Grund. Mat. 1, 97–108 (1955).
P. G. Odifreddi, ‘‘Strong reducibilities,’’ Bull. Am. Math. Soc. 4, 37–86 (1981).
P. G. Odifreddi, ‘‘Reducibilities,’’ in Handbook of Recursion Theory, Ed. by E. R. Griffor (North-Holland, Amsterdam, 1999), pp. 89–120.
E. L. Post, ‘‘Recursively enumerable sets of positive integers and their decision problems,’’ Bull. Am. Math. Soc. 50, 284–316 (1944).
R. I. Soare, Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Series Perspectives in Mathematical Logic (Springer, Berlin, 1987).
P. R. Young, ‘‘Linear orderings under one-one reducibility,’’ J. Symb. Logic 31, 70–85 (1966).
Funding
This work is supported by the Russian Science Foundation (project no. 18-11-00028). The author also thanks M.M. Yamaleev for drawing his attention to this problem.
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(Submitted by M. M. Arslanov)
