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Not every finite lattice is embeddable in the recursively enumerable degrees

A.H Lachlan, R.I Soare
1980 Advances in Mathematics  
An immediate consequence is that an embedding of dp as a lattice into the r.e. degrees with 00 going to b would simultaneously be an embedding into the b-r.e. degrees.  ...  From the latter by relativization for any r.e. degreeb there is no embedding of 9 as a lattice into the b-r.e. degrees with 02 being mapped to b. From [2, p. 5681 we have: LEMMA 1.  ... 
doi:10.1016/0001-8708(80)90027-4 fatcat:2vwdewhs2zf5hjsm4dfcasrhqa

Page 4301 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
The question of which lattices can be embedded into the r.e. degree structure R is one of the most important questions in recursion theory.  ...  be embedded into R.  ... 

Page 1385 of Mathematical Reviews Vol. , Issue 86d [page]

1986 Mathematical Reviews  
be shown that every finite lattice can be emledded (as a lattice) into R, the partially ordered set of recursively enumerable degrees.  ...  This is done by establishing a general embedding theorem for a large class of lattices that generalizes the embedding of the pentagon.  ... 

Lattice embeddings below a nonlow2 recursively enumerable degree

Rod Downey, Richard A. Shore
1996 Israel Journal of Mathematics  
We introduce techniques that allow us to embed below an arbitary nonlow 2 recursively enumerable degree any lattice currently known to be embedable into the recursively enumerable degrees.  ...  In the present paper we introduce a new technique that enables one to embed lattices into the recursively enumerable degrees below an arbitrary recursively enumerable nonlow 2 degree.  ...  embedded as lattices into R?  ... 
doi:10.1007/bf02762706 fatcat:b5tij4dkpral5m4p4xcvinrnqy

Preface

Richard A. Shore
1997 Annals of Pure and Applied Logic  
in a series of papers the latest of which appears here as "Possible degrees in recursive copies II", C.  ...  Symbolic Logic as "Decidability of the two quantifier theory in the recursively enumerable weak truth-table degrees and other distributive upper semi-lattices", K. Ambos-Spies, P. Fejer, S.  ...  be embedded into the enumerable Turing degrees", S.  ... 
doi:10.1016/s0168-0072(96)00025-5 fatcat:lsa3nc2ay5dxnekue2zf6bf3na

Computability, enumerability, unsolvability: Directions in recursion theory

1996 Computers and Mathematics with Applications  
Part II. Data in the computer. 2. Number systems. 3. Data formats. 4. Representing integer data. 5. Floating point numbers. Part III. Computer architecture and hardware operation. 6.  ...  The software component. 13. Operating systems--An overview. 14. The user view of operating systems. 15. The internal operating system. 16. File management. 17. Programming tools. 18.  ...  On the V3-theory of the factor lattice by the major subset relation (E. Herrmann). Degrees of generic sets (M. Kumabe). Embeddings into the recursively enumerable degrees (M. Lerman).  ... 
doi:10.1016/s0898-1221(96)90223-0 fatcat:dvj5uaqtwngedgvptajwlh7lmi

Page 4452 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
In the paper under review the authors show that for every nonzero recursively enumerable degree a, every countable distribu- tive lattice can be embedded preserving | into the semilattice of all r.e. degrees  ...  Miinchen, Munich, 1980; Zbl 484:03023] proved that every countable dis- tributive lattice can be embedded into the re. degrees preserving 1.  ... 

CGI programming on the world wide web

1996 Computers and Mathematics with Applications  
Part II. Data in the computer. 2. Number systems. 3. Data formats. 4. Representing integer data. 5. Floating point numbers. Part III. Computer architecture and hardware operation. 6.  ...  The software component. 13. Operating systems--An overview. 14. The user view of operating systems. 15. The internal operating system. 16. File management. 17. Programming tools. 18.  ...  On the V3-theory of the factor lattice by the major subset relation (E. Herrmann). Degrees of generic sets (M. Kumabe). Embeddings into the recursively enumerable degrees (M. Lerman).  ... 
doi:10.1016/s0898-1221(96)90221-7 fatcat:nvtygw2zavapvaounbg7kbswpy

Page 2530 of Mathematical Reviews Vol. , Issue 90E [page]

1990 Mathematical Reviews  
W- degrees between a and b) a lattice, and that the lattices that can be embedded into arbitrary W[a,b] with greatest element preserved are exactly the countable distributive lattices.  ...  (ii) For any degree of unsolvability a there exists a distributive lattice 2 = (L;<,N,U) which is recursively enumerable with respect to a such that d((L;<)) =a’.  ... 

The Recursively Enumerable Degrees [chapter]

Richard A. Shore
1999 Studies in Logic and the Foundations of Mathematics  
These results opened up a new chapter in the embedding problem for the r. e. degrees which has yet to be finished: Which lattices can be embedded into R (preserving the lattice structure, of course, but  ...  (Lachlan, Lerman, Thomason; see Soare [1987] IX.2) Every countable distributive lattice can be embedded into R as a lattice preserving 0.  ... 
doi:10.1016/s0049-237x(99)80022-6 fatcat:yviyyg7uhfbolegitvgof4ga7m

Page 1314 of Mathematical Reviews Vol. 50, Issue 5 [page]

1975 Mathematical Reviews  
50 £49560-9565 16 (1966), 537-569; MR 34 #4126] that if a and b are recursively enumerable degrees that join nontrivially to 0’, then there exists a nonzero recursively enumerable degree ¢ such that c<  ...  enumerable many-one degrees.  ... 

An extension of the recursively enumerable Turing degrees

Stephen G. Simpson
2007 Journal of the London Mathematical Society  
Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees.  ...  We exhibit a natural embedding of RT into Pw which is one-toone, preserves the semilattice structure of RT , carries 0 to 0, and carries 0 ′ to 1.  ...  Embedding R T into P w In this section we exhibit a specific, natural embedding of the countable upper semilattice R T into the countable distributive lattice P w . Definition 5.1.  ... 
doi:10.1112/jlms/jdl015 fatcat:gggbcm5sffb23axocnoioragmm

Page 4 of Mathematical Reviews Vol. , Issue 81K [page]

1981 Mathematical Reviews  
N. 81k:03039 tt-degrees of recursively enumerable Turing degrees. II. (Russian) Algebra i Logika 18 (1979), no. 4, 415-425, 507-508.  ...  An enumeration of the partial recursive functions P, is a total recursive function y: NP.  ... 

Page 1121 of Mathematical Reviews Vol. 52, Issue 4 [page]

1976 Mathematical Reviews  
The decision problem for recursively enumerable degrees. Bull. Amer. Math. Soc. 81 (1975), no. 6, 973-977.  ...  He defines the notions of recursive set, relative recursion, and the upper semi-lattice D of degrees of recursive unsolvability, citing results from the well- known papers of S. C. Kleene and E. L.  ... 

Page 6520 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews  
in the r.e. degrees, P’ is order-embedded in the degrees r.e. in and above 0’, these embeddings preserve least and greatest elements, and f corresponds to the operation under these embeddings.  ...  enumerable degrees and Post’s program.  ... 
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