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Page 1701 of Mathematical Reviews Vol. , Issue 87d
[page]

1987
*
Mathematical Reviews
*

A

*recursively**enumerable*set A is defined to be antimitotic if for all pairs*of*r.e. sets B,C with BUC = A and B,C disjoint,*the*infimum*of**the**degree**of*B and*the**degree**of*C equals*0*. ... Sacks demonstrated that every nonzero r.e.*degree**splits*, by showing that every nonrecursive r.e. set has a*splitting**into*r.e. sets*of*incomparable*degree*. ...##
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Page 638 of Mathematical Reviews Vol. , Issue 89B
[page]

1989
*
Mathematical Reviews
*

*of*

*splittings*

*of*V

*into*a direct sum, and d(V),

*the*Turing

*degree*

*of*V as a set? ... V. (1-WIL)

*Degrees*

*of*

*splittings*and bases

*of*

*recursively*

*enumerable*subspaces. Trans. Amer. Math. Soc. 302 (1987), no. 2, 683-714. ...

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An extended Lachlan splitting theorem

1996
*
Annals of Pure and Applied Logic
*

We show that

doi:10.1016/0168-0072(95)00039-9
fatcat:65jdqycxlfcezo2h4xld5geww4
*the*top*of*any diamond with bottom*0*in*the*r.e.*degrees*is also*the*top*of*a stack*of*n diamonds with bottom*0*. Let R be*the*upper semilattice*of**the**recursively**enumerable**degrees*. ... A rn~~irna~ pair consists*of*two incomparable r.e.*degrees*with in~mum equal to*the**recursive**degree**0*. An r.e.*degree*is cappable if it is one half*of*a minimal pair. ... Acknowledgements*The*first author was supported by a NSF grant DMS-9100114.*The*second author was partially supported by*the*National NSF*of*China.*The*second author wishes to thank Richard A. ...##
###
Friedberg splittings of recursively enumerable sets

1993
*
Annals of Pure and Applied Logic
*

Stob, Friedberg

doi:10.1016/0168-0072(93)90092-r
fatcat:v2eb3cqujfbqngqwgd6wltnqte
*splittings**of**recursively**enumerable*sets, Annals*of*Pure and Applied Logic 59 (1993) 175-199. ... In*the*present paper we continue our investigations, this time analyzing Friedberg*splittings*and in particular their orbits and*degrees*for various classes*of*r.e. sets. ... Let*0*be*the*orbit generated by*the*e-*splits**of*a creative set. Thus deg(*0*) 2 PS by (4.3). If C U D is an e-*split*, it is an f-*split*and so has promptly simple*degree*by (4.2) . ...##
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Page 3777 of Mathematical Reviews Vol. , Issue 85i
[page]

1985
*
Mathematical Reviews
*

A

*recursively**enumerable**degree*is said to be cappable if it is*0*or is part*of*some minimal pair*of*r.e.*degrees*. K. Ambos-Spies,*the*reviewer, R. A. Shore and R. I. ... From*the*text: “We prove*the*following assertions: (a) There exists a w-mitotic*recursively**enumerable*set*of**degree**0*’, not having a btt-mitotic decomposition. ...##
###
Two splitting theorems for beta-recursion theory

1980
*
Annals of Mathematical Logic
*

, was proved by Shore [14] where he showed how to

doi:10.1016/0003-4843(80)90015-7
fatcat:u4ralkm7cvgivlenvz5xbjyvzi
*split*a regular a-r.e, set A*into*two a-r.e, sets B and C whose join has*the*same ~x-*degree*as A, while*the*s-*degrees**of*B and C are both less than*the*...*The*regular sets theorem*of*Sacks [12] then enabled Shore to cJnclude that he could*split*sets in any non-a-*recursive*a-r,e,*degree*. ...*Degrees**of*sets with small complement In this section we want to show that, for weakly admissible /3,*the**splitting*theorem*of*Section 3*splits*sets in every t,r,e,*degree*,*The*t.r.e,*degrees*can be shown ...##
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Page 3030 of Mathematical Reviews Vol. , Issue 89F
[page]

1989
*
Mathematical Reviews
*

Given an r.e. subalgebra A, one says that A can be

*split**into*two r.e. subalgebras A’ and A” if A =*the*subalgebra generated by A’ U A" and A’'q A” = {*0*, 1}. ...*The*author shows that every nonrecursive r.e. subalgebra can be*split**into*two nonrecursive r.e. subalgebras. This result can be viewed as a Boolean analogue*of*Friedberg’s*splitting*theorem. ...##
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Π10 classes and minimal degrees

1997
*
Annals of Pure and Applied Logic
*

There is a non-empty III: class

doi:10.1016/s0168-0072(96)00029-2
fatcat:hjks4x3o2vhitj4nz3l4ngsfwq
*of*reals, each*of*which computes a real*of*minimal (Turing)*degree*. Corollary. WKL t "there is a minimal Turing*degree*". ... Working*recursively*in*0*', Sacks could not use Spector's division*into*cases, since those cases are Ci and ZI!. Instead*of**recursive*trees, Sacks used*recursively**enumerable*trees. ... In either case, again*the*greatest member*of**the*former*0*sequence that remains in M is at most*the*former a(j' -l), so we can argue as above that some x satisfying R(e) is*enumerated**into*W.*0*Before ...##
###
Embedding and Coding below a 1-Generic Degree

2003
*
Notre Dame Journal of Formal Logic
*

We show that

doi:10.1305/ndjfl/1091122498
fatcat:ck5jsjjuafdkppkprjgeldddqu
*the*theory*of*D( g), where g is a 2-generic or a 1-generic*degree*below*0*, interprets true first order arithmetic. ... To this end we show that 1-genericity is sufficient to find*the*parameters needed to code a set*of**degrees*using Slaman and Woodin's method*of*coding in Turing*degrees*. ...*The*first author was also supported by a fellowship from*the*Lady Davis Trust. ...##
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Page 27 of Mathematical Reviews Vol. , Issue 2000a
[page]

2000
*
Mathematical Reviews
*

*The*goal

*of*this paper is to show that above every incomplete wtt

*degree*a, there is an embedding

*of*Ms

*into*

*the*wtt

*degrees*

*of*

*the*d.r.e. sets wtt above a sending

*0*(in Ms) to a. ... For example,

*the*jump

*of*a

*recursively*

*enumerable*(r.e.) set is not itself re. (except in

*the*case

*of*a

*recursive*set), and a high

*degree*is not necessarily incomplete as

*the*authors claim. Sh. T. ...

##
###
Page 818 of Mathematical Reviews Vol. , Issue 81C
[page]

1981
*
Mathematical Reviews
*

Then he proves

*the*characteri- | zation theorem: A regular, non-a-*recursive*set G lying below*0*is*of*minimal a-*degree*if and only if it is generic. ... D. 81¢c:03035 Structure*of**the*upper semilattice*of**recursively**enumerable*m-*degrees*and related questions. I. (Russian) Algebra i Logika 17 (1978), no. 6, 643-683, 746. ...##
###
Page 1271 of Mathematical Reviews Vol. , Issue 95c
[page]

1995
*
Mathematical Reviews
*

*into*

*the*r.e.

*degrees*preserving

*0*and 1. ... It is a very important and difficult problem in

*recursion*theory to characterize

*the*embeddability

*of*finite lattices in

*the*

*recursively*

*enumerable*(r.e.)

*degrees*. ...

##
###
members of thin Π_1^0 classes and generic degrees
[article]

2020
*
arXiv
*
pre-print

*enumerable*, and can be minimal

*degree*below

*0*'. ... In 1993, Cenzer, Downey, Jockusch and Shore initiated

*the*study

*of*Turing

*degrees*

*of*members

*of*thin Π^0_1 classes, and proved that

*degrees*containing no members

*of*thin Π^0_1 classes can be

*recursively*... Thus, we obtain a path C in a thin class [T ] with C ≡ T A, and A is not

*of*thin-free

*degree*. ✷ ...

##
###
Automorphisms of the lattice of recursively enumerable sets: Orbits

1992
*
Advances in Mathematics
*

*Enumerate*x (

*the*x

*of*step 1)

*into*Ct+r. ... Suppose that there is a simultaneous

*enumeration*

*of*a

*recursive*array including all

*the*above such that A \ p,, =

*0*= B \ o,,, for all n. ...

##
###
Splitting an $\alpha $-recursively enumerable set

1975
*
Transactions of the American Mathematical Society
*

*The*result is also strengthened to apply to <ca, and various corollaries about

*the*structure

*of*

*the*a and ca

*recursively*

*enumerable*

*degrees*are proved. ... Thus for example one should mark

*the*difference between

*the*arguments for

*the*Friedburg-Muchnik solution

*of*Post's problem [2] and Sacks'

*splitting*theorem [3], [7] . ... On

*the*other hand AUB = C clearly implies that C ^ A V B. D We can now prove

*the*usual corollaries

*of*

*the*

*splitting*theorem. Let c and d be a-r.e.

*degrees*such that d is not a-

*recursive*. ...

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