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An extended Lachlan splitting theorem

1996
*
Annals of Pure and Applied Logic
*

For any n > 1, a is the top of

doi:10.1016/0168-0072(95)00039-9
fatcat:65jdqycxlfcezo2h4xld5geww4
*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. ...##
###
Recursively enumerable sets and degrees

1978
*
Bulletin of the American Mathematical Society
*

Basic facts and

doi:10.1090/s0002-9904-1978-14552-2
fatcat:arxp4btvhzfzjeoufybmnemt2u
*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]). ...##
###
The Recursively Enumerable Degrees
[chapter]

1999
*
Studies in Logic and the Foundations of Mathematics
*

There are then embeddings of P which cannot be

doi:10.1016/s0049-237x(99)80022-6
fatcat:yviyyg7uhfbolegitvgof4ga7m
*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. ...##
###
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*an*unpublished proof by Shelah of the above*theorem*of*Lachlan*.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

1995
*
Proceedings of the National Academy of Sciences of the United States of America
*

Many of the most significant

doi:10.1073/pnas.92.2.617
pmid:11607508
pmcid:PMC42793
fatcat:4a6nozmejjdlblqaee3wz2diwq
*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*. ...##
###
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. ...

##
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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. ...##
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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

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. ...

##
###
Degrees of Unsolvability
[chapter]

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 ...

##
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Determining automorphisms of the recursively enumerable sets

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

doi:10.1090/s0002-9939-1977-0446931-5
fatcat:o3q4gbdsoffqtpwulpn66pskwq
*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] ). ...##
###
Definability in the Recursively Enumerable Degrees

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 ...

##
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Definable Encodings in the Computably Enumerable Sets

2000
*
Bulletin of Symbolic Logic
*

What these results have in common is that the guts of the proofs of these

doi:10.2307/421206
fatcat:oekr3krowrhy7gfo62tpxpvahi
*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] . ...##
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Determining Automorphisms of the Recursively Enumerable Sets

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. ...

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