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$$\Pi^1_1$$ relations and paths through $$\mathcal O$$. (English) Zbl 1107.03051
First, the authors give a syntactical characterization of intrinsically $$\Pi_1^1$$-relations on structures. Then they study such relations on structures of maximal Scott rank. For the most natural examples like superatomic parts of Boolean algebras, elements of $$p$$-groups possessing height, and well-ordered segments of Harrison orderings, they find out that the set of possible Turing degrees of such subsets is the same as the set $$\mathcal P$$ of Turing degrees of full paths through Kleene’s system $$\mathcal O$$. By this, the structure of $$\mathcal P$$ is especially interesting to study. In particular, it is shown that there is a path in $$\mathcal O$$ in which no noncomputable hyperarithmetical part is computable, that $$\mathcal P$$ contains a minimal pair, and that every countable distributive lattice could be embedded in $$\mathcal P$$ and $$\mathcal P(\leq {\mathbf c})$$, for each $${\mathbf 0}'\leq {\mathbf c}\in{\mathcal P}$$.

##### MSC:
 03D45 Theory of numerations, effectively presented structures 03D28 Other Turing degree structures 03F15 Recursive ordinals and ordinal notations
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##### References:
 [1] DOI: 10.1016/S0168-0072(97)00059-6 · Zbl 0927.03072 [2] Infinitistic methods pp 103– (1959) [3] DOI: 10.2140/pjm.1969.30.67 · Zbl 0181.30602 [4] Computability, enumerability, unsolvability 224 pp 93– (1996) [5] Effective model theory vs. recursive model theory 55 pp 1168– (1990) [6] Infinitary logic and admissible sets 34 pp 226– (1969) [7] DOI: 10.1016/0168-0072(88)90014-0 · Zbl 0651.03034 [8] DOI: 10.1016/0003-4843(78)90026-8 · Zbl 0404.03020 [9] Aspects of effective algebra pp 26– (1981) [10] Recursive well-orderings 20 pp 151– (1955) [11] DOI: 10.1016/0168-0072(89)90015-8 · Zbl 0678.03012 [12] DOI: 10.1002/malq.19960420110 · Zbl 0845.03021 [13] Computable structures and the hyperarithmetical hierarchy (2000) · Zbl 0960.03001 [14] DOI: 10.1002/malq.19960420139 · Zbl 0859.03016 [15] DOI: 10.1016/S0168-0072(96)00026-7 · Zbl 0877.03022 [16] The theory of models pp 329– (1965) [17] DOI: 10.1016/0168-0072(94)00043-3 · Zbl 0837.03036 [18] Higher recursion theory (1990) · Zbl 0716.03043 [19] Southeast Asian Conference on Logic (Singapore, 1981) pp 185– (1983) [20] Fundamenta Mathematical 87 pp 161– (1975) [21] DOI: 10.1016/S0168-0072(01)00087-2 · Zbl 1016.03034 [22] DOI: 10.1090/S0002-9947-1968-0244049-7 [23] DOI: 10.1016/0168-0072(91)90097-6 · Zbl 0756.03022 [24] DOI: 10.1023/A:1021758312697 [25] DOI: 10.1007/BF01669456 · Zbl 0407.03040 [26] Proceedings of the American Mathematical Society 54 pp 311– (1976) [27] One hundred and two problems in mathematical logic 40 pp 113– (1975) [28] Theory of recursive functions and effective computability (1967) · Zbl 0183.01401 [29] DOI: 10.1016/0003-4843(77)90009-2 · Zbl 0376.02032 [30] Proceedings of the American Mathematical Society 39 pp 178– (1973) [31] Degrees of unsolvability (1983) · Zbl 0542.03023 [32] DOI: 10.4153/CJM-1971-024-7 · Zbl 0272.02064 [33] Infinite abelian groups (1954)
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