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**1 - 2**of**2**### RESTRICTED JUMP INTERPOLATION IN THE D.C.E. DEGREES

, 2009

"... It is shown that for any 2-computably enumerable Turing degree l, any computably enumerable degree a, and any Turing degree s, if l ′ = 0 ′, l < a, s ≥ 0 ′ , and s is c.e. in a, then there is a 2-computably enumerable degree x with the following properties: (1) l < x < a, and (2) x ′ = s. ..."

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It is shown that for any 2-computably enumerable Turing degree l, any computably enumerable degree a, and any Turing degree s, if l ′ = 0 ′, l < a, s ≥ 0 ′ , and s is c.e. in a, then there is a 2-computably enumerable degree x with the following properties: (1) l < x < a, and (2) x ′ = s. 1

### unknown title

, 2007

"... Abstract A set A ` ! is called computably enumerable (c.e., for short), if there is an algorithmto enumerate the elements of it. For sets A, B `!, we say that A is bounded Turingreducible to B if there is a Turing functional, \Phi say, with a computable bound of oraclequery bits such that A is compu ..."

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Abstract A set A ` ! is called computably enumerable (c.e., for short), if there is an algorithmto enumerate the elements of it. For sets A, B `!, we say that A is bounded Turingreducible to B if there is a Turing functional, \Phi say, with a computable bound of oraclequery bits such that A is computed by \Phi equipped with an oracle B, written A <=bT B.Let E bT be the structure of the c.e. bT-degrees, the c.e. degrees under the boundedTuring reductions. In this paper we study the continuity properties in E bT. We showthat for any c.e. bT-degree b 6 = 0, 00, there is a c.e. bT-degree a> b such that for anyc.e. bT-degree x, b ^ x = 0 if and only if a ^ x = 0. This is the first continuity property