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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 2271 of Mathematical Reviews Vol. , Issue 87e [page]

1987 Mathematical Reviews  
(D-DORT); 87e:03094 Lerman, M. (1-CT) Lattice embeddings into the recursively enumerable degrees. J. Symbolic Logic 51 (1986), no. 2, 257-272.  ...  The authors give a condition, the nonembeddability condition (NEC), which implies that a finite lattice cannot be embedded in the upper semilattice of the recursively enumerable (r.e.)  ... 

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.  ... 

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

2000 Mathematical Reviews  
finite ranked partial lattice into the c.e. degrees.  ...  The author introduces constructions for embedding a given partial lattice into the poset # of computably enumerable (c.e.) degrees (the language consists of a single nonlogical symbol, <), and also for  ... 

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

Carl G. Jockusch, Jeanleah Mohrherr
1985 Proceedings of the American Mathematical Society  
It is shown that the four element Boolean algebra can be embedded in the recursively enumerable truth-table degrees with least and greatest elements preserved.  ...  We show the existence of two incomparable recursively enumerable (r.e.) tt-degrees with supremum 0' (the highest r.e. tt-degree) and infimum 0 (the lowest).  ...  The next result shows that the modular five-element nondistributive lattice known as 1-3-1 can be embedded in the r.e. truth-table degrees. Theorem 2.  ... 
doi:10.1090/s0002-9939-1985-0781069-3 fatcat:qwodvdpzznbvdptmvzptsjax5m

Page 3578 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews  
Turing degrees can be embedded into the noncomplete £3-enumeration degrees, it fol- lows by branching the top element of Ms that the lattice Sg can be embedded into the £5-enumeration degrees.  ...  This result is inter- esting to compare with the c.e. Turing degrees case: it is known that the lattice Ss cannot be embedded into the c.e. Turing de- grees. Marat M.  ... 

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 1367 of Mathematical Reviews Vol. , Issue 90C [page]

1990 Mathematical Reviews  
The result is then extended from embeddings into the degrees < 0’ to embeddings into the degrees < a, where a is an arbitrary non-GL» degree, i.e. any degree which satisfies a” > (aU0’)’. C. G.  ...  In the paper under review, this result of Cooper’s is extended to show that any recursively presented lattice with distinct least and greatest elements can be embedded in the degrees < 0’, with the embedding  ... 

Page 2446 of Mathematical Reviews Vol. , Issue 89E [page]

1989 Mathematical Reviews  
can be embedded into the polynomial Turing degrees.  ...  O 89e:03070 03D30 Han, Xiao Feng (PRC-BJ) On the embedding of distributive lattices into recursive polynomial degrees. Acta Math. Sinica (N.S.) 3 (1987), no. 2, 97-102.  ... 

Page 4722 of Mathematical Reviews Vol. , Issue 87i [page]

1987 Mathematical Reviews  
“To summarize the rest of the paper, in Section 2 we discuss embedding lattices and other partially ordered sets into the r.e. tt-degrees.  ...  {g,} and {W,} are standard enumerations of the partial recursive functions and the recursively enumerable sets, respectively).  ... 

Page 3145 of Mathematical Reviews Vol. , Issue 2000e [page]

2000 Mathematical Reviews  
Summary: “The paper is concerned with the construction of in- tervals of computably enumerable degrees in which the lattice Ms cannot be embedded.  ...  Summary: “Lattice representations are an important tool for com- putability theorists when they embed nondistributive lattices into degree-theoretic structures.  ... 

Conjectures and questions from Gerald Sacks's Degrees of Unsolvability

Richard A. Shore
1997 Archive for Mathematical Logic  
We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the  ...  Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching.  ...  (C3) There exist two recursively enumerable degrees with no greatest lower bound in the upper semi-lattice of recursively enumerable degrees.  ... 
doi:10.1007/s001530050063 fatcat:wmp32iusbvffnbkv6rsosghtoq

Page 5307 of Mathematical Reviews Vol. , Issue 85m [page]

1985 Mathematical Reviews  
Kobzev (Tbilisi) Yang, Dong Ping (1-CRNL) 85m:03032 On the embedding of a-recursive presentable lattices into the a-recursive degrees below 0’. J. Symbolic Logic 49 (1984), no. 2, 488-502. S. C.  ...  ., 1963; MR 32 #4013] extended this result to show that in fact such an ordering may be embedded in the recursively enumerable (r.e.) T-degrees. A. H. Lachlan and R. Lebeuf [J.  ... 

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.  ... 
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