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An extended Lachlan splitting theorem

Steffen Lempp, Sui Yuefei
1996 Annals of Pure and Applied Logic  
For any n > 1, a is the top of an n-diamond if there is a nontrivial splitting ao and al of a such that the infimum of a0 and al exists and is the top of an (pt -I)-diamond.  ...  Lachlan [3] and Yates [7] proved that there is a minimal pair in R. ~bos-Spies et al.  ...  Theorem 1.  ... 
doi:10.1016/0168-0072(95)00039-9 fatcat:65jdqycxlfcezo2h4xld5geww4

Recursively enumerable sets and degrees

Robert I. Soare
1978 Bulletin of the American Mathematical Society  
Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11.  ...  Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R. Introduction. G. E.  ...  Certain "splitting" theorems arose early in the subject and have played an important role in questions of both structure and decidability of &. THEOREM 6.2 (FRIEDBERG SPLITTING THEOREM [Fr4]).  ... 
doi:10.1090/s0002-9904-1978-14552-2 fatcat:arxp4btvhzfzjeoufybmnemt2u

The Recursively Enumerable Degrees [chapter]

Richard A. Shore
1999 Studies in Logic and the Foundations of Mathematics  
There are then embeddings of P which cannot be extended to ones of Q 0 by the minimal pair theorem (Theorem 2.6) and there are ones which cannot be extended to Q 1 by the Sacks splitting theorem (Theorem  ...  These techniques were first introduced by Lachlan to prove that the Sacks Splitting and Density Theorems cannot be combined. Theorem 2.15.  ... 
doi:10.1016/s0049-237x(99)80022-6 fatcat:yviyyg7uhfbolegitvgof4ga7m

Page 1689 of Mathematical Reviews Vol. 55, Issue 6 [page]

1978 Mathematical Reviews  
According to Harnik his proof was inspired by an unpublished proof by Shelah of the above theorem of Lachlan. The author also derives several consequences of this lemma.  ...  Lachlan [Fund.  ... 

Algebraic aspects of the computably enumerable degrees

T. A. Slaman, R. I. Soare
1995 Proceedings of the National Academy of Sciences of the United States of America  
Many of the most significant theorems giving an algebraic insight into Rk have asserted either extension or nonextension of embeddings.  ...  to an embedding of Q into Rt.  ...  the minimal pair and splitting theorems.  ... 
doi:10.1073/pnas.92.2.617 pmid:11607508 pmcid:PMC42793 fatcat:4a6nozmejjdlblqaee3wz2diwq

Page 21 of Mathematical Reviews Vol. , Issue 84a [page]

1984 Mathematical Reviews  
The latter result is obtained as a corollary of a strengthening of the Robinson splitting theorem and the answer to the generalised nondiamond question (found by J. R. Shoenfield, R. I.  ...  Let » denote an asymptotically optimal programming language which gives an enumeration 9,,9,... of all partial recursive functions for which a special variant of the iteration theorem holds.  ... 

Page 891 of Mathematical Reviews Vol. , Issue 84c [page]

1984 Mathematical Reviews  
Extendibility of an embedding to an isomorphism plays an important role, which is proved by combining the compactness theorem and algebra.  ...  A universal theory T of commutative rings (possibly in an extended language) is here said to be special if every ring of the form R[x]/(ax) (a€R,RFET) is a model of T.  ... 

Page 1876 of Mathematical Reviews Vol. , Issue 89D [page]

1989 Mathematical Reviews  
{Reviewer's remark: The result is similar to the Lachlan non- splitting theorem [A. H. Lachlan, Ann. Math. Logic 9 (1976), no. 4, 307-365; MR 53 #12912].}  ...  Rice’s theorem and the Rice-Shapiro theorem give complete char- acterizations of the recursive and r.e. index sets.  ... 

Page 27 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
Since on the other side there is an announced result of Cooper, Lachlan and Slaman that every nonlow c.e. set has a proper splitting with one splitting half not low, the above-stated splitting property  ...  In this paper an open problem in splitting theory is answered (in a strong way).  ... 

Embedding the diamond lattice in the recursively enumerable truth-table degrees

Carl G. Jockusch, Jeanleah Mohrherr
1985 Proceedings of the American Mathematical Society  
Lachlan proved in his well-known "nondiamond theorem" [5, Theorem 5] that the diamond lattice cannot be embedded in the r.e. Turing degrees with 0 and 1 preserved.  ...  His construction involved splitting a creative set K into two disjoint r.e. -  ...  Theorem 2 extends in an obvious way from triples to «-tuples so that the 1-zz-l lattice is embeddable in the r.e. tt-degrees with 0 and 1 preserved.  ... 
doi:10.1090/s0002-9939-1985-0781069-3 fatcat:qwodvdpzznbvdptmvzptsjax5m

Degrees of Unsolvability [chapter]

Klaus Ambos-Spies, Peter A. Fejer
2014 Handbook of the History of Logic  
Lachlan [Lachlan, 1972] and Lerman (unpublished) extended this to the countable case.  ...  For the above definition of the arithmetical degrees a join theorem, i.e., an extended version of this completeness theorem including joins is proven for the case of ω-c.e. operators: For such an operator  ... 
doi:10.1016/b978-0-444-51624-4.50010-1 fatcat:clf7varewrh6lc4jq2c2sp6pgy

Determining automorphisms of the recursively enumerable sets

Richard A. Shore
1977 Proceedings of the American Mathematical Society  
If G* is the class of recursive sets modulo finite sets or 911* C 6* C S * (91L* = maximal sets, S * = simple sets) then there is an automorphism of (the lattice generated by) G* which does not extend  ...  Theorem. If G* is any nontrivial recursively invariant subclass of S *, then any automorphism of S* is determined uniquely by its action on Q*. THEOREM.  ...  Theorem 8. There is an automorphism a? of 91* which does not extend to one of&*. Proof. Let A be an /--maximal set with a maximal superset and B one without any (one exists by Lachlan [1968] ).  ... 
doi:10.1090/s0002-9939-1977-0446931-5 fatcat:o3q4gbdsoffqtpwulpn66pskwq

Definability in the Recursively Enumerable Degrees

André Nies, Richard A. Shore, Theodore A. Slaman
1996 Bulletin of Symbolic Logic  
Theorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degree a there are r.e. degrees b, c < a such that b ∨ c = a. Theorem 1.3 (Sacks Density Theorem [1964]).  ...  For every pair of nonrecursive r.e. degrees a < b there is an r.e. degree c such that a < c < b.  ...  , (by the Sacks splitting theorem) and noncuppable degrees (Lachlan [1966a] ); • degrees which split over every smaller degree (any low degree a, i. e., a = 0 , by Robinson [1971] , any low 2 degree  ... 
doi:10.2307/421171 fatcat:sot7revil5gshppsw4op7faqeu

Definable Encodings in the Computably Enumerable Sets

Peter A. Cholak, Leo A. Harrington
2000 Bulletin of Symbolic Logic  
What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.  ...  definable quotient structure of ε since "Χ is finite" is definable in ε; "Χ is finite" iff all subsets of Χ are computable (it takes a little computability theory to show if Χ is infinite then Χ has an  ...  This result was later extended in Herrmann [1989] .  ... 
doi:10.2307/421206 fatcat:oekr3krowrhy7gfo62tpxpvahi

Determining Automorphisms of the Recursively Enumerable Sets

Richard A. Shore
1977 Proceedings of the American Mathematical Society  
Theorem 8. There is an automorphism a? of 91* which does not extend to one of&*. Proof. Let A be an /--maximal set with a maximal superset and B one without any (one exists by Lachlan [1968] ).  ...  In trying to find an automorphism of 91 * not extending to one of S *, we (R.I.  ... 
doi:10.2307/2041915 fatcat:cr3vvrjeqnazhbasyz4nfrom3e
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