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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.  ... 

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.  ... 

An extended Lachlan splitting theorem

Steffen Lempp, Sui Yuefei
1996 Annals of Pure and Applied Logic  
We show that 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.  ... 
doi:10.1016/0168-0072(95)00039-9 fatcat:65jdqycxlfcezo2h4xld5geww4

Friedberg splittings of recursively enumerable sets

Rod Downey, Michael Stob
1993 Annals of Pure and Applied Logic  
Stob, Friedberg 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) .  ... 
doi:10.1016/0168-0072(93)90092-r fatcat:v2eb3cqujfbqngqwgd6wltnqte

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

Steven Homer
1980 Annals of Mathematical Logic  
, was proved by Shore [14] where he showed how to 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  ... 
doi:10.1016/0003-4843(80)90015-7 fatcat:u4ralkm7cvgivlenvz5xbjyvzi

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.  ... 

Π10 classes and minimal degrees

Marcia J. Groszek, Theodore A. Slaman
1997 Annals of Pure and Applied Logic  
There is a non-empty III: class 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  ... 
doi:10.1016/s0168-0072(96)00029-2 fatcat:hjks4x3o2vhitj4nz3l4ngsfwq

Embedding and Coding below a 1-Generic Degree

Antonio Montalb�an, Noam Greenberg
2003 Notre Dame Journal of Formal Logic  
We show that 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.  ... 
doi:10.1305/ndjfl/1091122498 fatcat:ck5jsjjuafdkppkprjgeldddqu

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]

Frank Stephan, Guohua Wu, Bowen Yuan
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. ✷  ... 
arXiv:2008.04539v1 fatcat:t34sa4hqzfdj3hycwg4zcwrr7y

Automorphisms of the lattice of recursively enumerable sets: Orbits

R.G Downey, Michael Stob
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.  ... 
doi:10.1016/0001-8708(92)90065-s fatcat:53a7j4mdubgpnf5cidhlxp2p7m

Splitting an $\alpha $-recursively enumerable set

Richard A. Shore
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.  ... 
doi:10.1090/s0002-9947-1975-0379154-1 fatcat:sdrkyywqpvbkzle4e3edhi34gu
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