S. Yu. Podzorov in [Mat. Trudy, 9, No. 2, 109-132 (2006)] proved the following theorem. Let 〈L, ≤ L 〉 be a local lattice and v a numbering of L such that the relation v(x) ≤ L v(y) is Δ0 2-computable. Then there is a numbering μ of L such that the relation μ(x) ≤ L μ(y) is computably enumerable. Podzorov also asked whether the hypothesis that 〈L, ≤ L 〉 is a local lattice is needed or the theorem is true of any partially ordered set (poset). We answer his question by constructing a poset for which the conclusion of the theorem fails.
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References
S. Yu. Podzorov, “Enumerated distributive semilattices,” Mat. Trudy, 9, No. 2, 109–132 (2006).
D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko, “Degree spectra and computable dimensions in algebraic structures,” Ann. Pure Appl. Log., 115, Nos. 1-3, 71–113 (2002).
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Translated from Algebra i Logika, Vol. 51, No. 4, pp. 423-428, July-August, 2012.
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J. Wallbaum. A Δ0 2-poset with no positive presentation. Algebra Logic 51, 281–284 (2012). https://doi.org/10.1007/s10469-012-9191-8
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DOI: https://doi.org/10.1007/s10469-012-9191-8