In the study of cappable and noncappable properties of the recursively enumerable (r.e.) degrees, Lempp suggested a conjecture which asserts that for all r.e. degrees a and b, if a ≮ b then there exists an r.e. degr...In the study of cappable and noncappable properties of the recursively enumerable (r.e.) degrees, Lempp suggested a conjecture which asserts that for all r.e. degrees a and b, if a ≮ b then there exists an r.e. degree c such that c ≮ a and c ≮ b and c is cappable. We shall prove in this paper that this conjecture holds under the condition that a is high. Working below a high r.e. degree h, we show that for any r.e. degree b with h ≮ b, there exist r.e. degrees aO and al such that a0, al ≮ b, aO,a1 ≮ h, and aO and a1 form a minimal pair.展开更多
Given any [c],[a],[d]∈R/M such that [d]≤[a]≤[c], [a] is locally noncuppable between [c] and [d] if [d]<[a] ≤[c]and [a] V [b] < [c] for any [b]∈R/M such that [d]≤ [ b ] < [ c ]. It will be shown that giv...Given any [c],[a],[d]∈R/M such that [d]≤[a]≤[c], [a] is locally noncuppable between [c] and [d] if [d]<[a] ≤[c]and [a] V [b] < [c] for any [b]∈R/M such that [d]≤ [ b ] < [ c ]. It will be shown that given any nonzero [ c ] ∈ R/M, there are [ a ], [ d ]∈ R/M such that [d]<[a]≤[c] and[a] is locally noncuppable between [ c ] and[d].展开更多
基金This reserch is supported by the National Natural Science Foundation of China (No.19971090).
文摘In the study of cappable and noncappable properties of the recursively enumerable (r.e.) degrees, Lempp suggested a conjecture which asserts that for all r.e. degrees a and b, if a ≮ b then there exists an r.e. degree c such that c ≮ a and c ≮ b and c is cappable. We shall prove in this paper that this conjecture holds under the condition that a is high. Working below a high r.e. degree h, we show that for any r.e. degree b with h ≮ b, there exist r.e. degrees aO and al such that a0, al ≮ b, aO,a1 ≮ h, and aO and a1 form a minimal pair.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 19971090).
文摘Given any [c],[a],[d]∈R/M such that [d]≤[a]≤[c], [a] is locally noncuppable between [c] and [d] if [d]<[a] ≤[c]and [a] V [b] < [c] for any [b]∈R/M such that [d]≤ [ b ] < [ c ]. It will be shown that given any nonzero [ c ] ∈ R/M, there are [ a ], [ d ]∈ R/M such that [d]<[a]≤[c] and[a] is locally noncuppable between [ c ] and[d].
