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Hristo Ganchev [8]Hristo A. Ganchev [2]
  1.  17
    A structural dichotomy in the enumeration degrees.Hristo A. Ganchev, Iskander Sh Kalimullin, Joseph S. Miller & Mariya I. Soskova - 2022 - Journal of Symbolic Logic 87 (2):527-544.
    We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half of a nontrivial relativized $\mathcal {K}$ -pair. This leads to a structural dichotomy in the enumeration degrees.
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  2.  29
    Initial segments of the enumeration degrees.Hristo Ganchev & Andrea Sorbi - 2016 - Journal of Symbolic Logic 81 (1):316-325.
    Using properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be (...)
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  3.  11
    The jump operator on the ω-enumeration degrees.Hristo Ganchev & Ivan N. Soskov - 2009 - Annals of Pure and Applied Logic 160 (3):289-301.
    The jump operator on the ω-enumeration degrees was introduced in [I.N. Soskov, The ω-enumeration degrees, J. Logic Computat. 17 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of and of the enumeration degrees are isomorphic. In the second part of the paper we (...)
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  4.  11
    A structural dichotomy in the enumeration degrees.Hristo A. Ganchev, Iskander Sh Kalimullin, Joseph S. Miller & Mariya I. Soskova - 2020 - Journal of Symbolic Logic:1-18.
    We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half a nontrivial relativized K-pair. This leads to a structural dichotomy in the enumeration degrees.
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  5.  14
    Interpreting true arithmetic in the local structure of the enumeration degrees.Hristo Ganchev & Mariya Soskova - 2012 - Journal of Symbolic Logic 77 (4):1184-1194.
    We show that the theory of the local structure of the enumeration degrees is computably isomorphic to the theory of first order arithmetic. We introduce a novel coding method, using the notion of a K-pair, to code a large class of countable relations.
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  6.  22
    Cupping and definability in the local structure of the enumeration degrees.Hristo Ganchev & Mariya I. Soskova - 2012 - Journal of Symbolic Logic 77 (1):133-158.
    We show that every splitting of ${0}_{\mathrm{e}}^{\prime }$ in the local structure of the enumeration degrees, $$\mathcal{G}_{e} , contains at least one low-cuppable member. We apply this new structural property to show that the classes of all $\mathcal{K}$ -pairs in $\mathcal{G}_{e}$ , all downwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees and all upwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees are first order definable in $\mathcal{G}_{e}$.
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  7.  50
    A note on the enumeration degrees of 1-generic sets.Liliana Badillo, Caterina Bianchini, Hristo Ganchev, Thomas F. Kent & Andrea Sorbi - 2016 - Archive for Mathematical Logic 55 (3):405-414.
    We show that every nonzero $${\Delta^{0}_{2}}$$ enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper.
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  8.  13
    The automorphism group and definability of the jump operator in the $$\omega $$ ω -enumeration degrees.Hristo Ganchev & Andrey C. Sariev - 2021 - Archive for Mathematical Logic 60 (7):909-925.
    In the present paper, we show the first-order definability of the jump operator in the upper semi-lattice of the \-enumeration degrees. As a consequence, we derive the isomorphicity of the automorphism groups of the enumeration and the \-enumeration degrees.
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  9.  16
    The high/low hierarchy in the local structure of the image-enumeration degrees.Hristo Ganchev & Mariya Soskova - 2012 - Annals of Pure and Applied Logic 163 (5):547-566.
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