Vol. 15, No. 4, 1965

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On relative coimmunity

Thomas Graham McLaughlin

Vol. 15 (1965), No. 4, 1319–1327

The paper relates to questions raised by A. A. Muchnik in a 1956 Doklady abstract, namely, whether a noncreative r.e. set can be simple in a creative one, and whether a creative r.e. set can be simple in a noncreative one. We furnish a negative answer to the second question, and give a variety of partial results having to do with the first. Thus, we show that no universal set can have immune relative complement inside a noncreative r.e. set and that any r.e. set which is hyperhypersimple in a creative set must itself be creative; whereas, there exist three sets α, β, γ, α β γ, such that β is creative, α and γ are nonuniversal, and both β α and γ β are hyperhyperimmune.

In addition, we answer two questions of J. P. Cleave regarding the comparison of effectively inseparable (e.i.) and “almost effectively inseparable” (almost e.i.) sequences of r.e. sets. Thus: a sequence can be almost e.i. without being e.i.; and an almost e.i. sequence of disjoint r.e. sets may have a noncreative union.

Mathematical Subject Classification
Primary: 02.70
Received: 6 June 1963
Revised: 16 July 1964
Published: 1 December 1965
Thomas Graham McLaughlin