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Embedding jump upper semilattices into the Turing degrees

Antonio Montalbán
2003 Journal of Symbolic Logic (JSL)  
we call jump, and is the jusl of Turing degrees.  ...  We prove that every countable jump upper semilattice can be embedded in , where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that  ...  Observe that, D = D, ≤ T , ∨, , the set of Turing degrees together with the Turing reduction, the join operation and the Turing Jump is a jusl.  ... 
doi:10.2178/jsl/1058448451 fatcat:lvp5qgexibezvhzanwolzybwvm

Page 5961 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
5961 03D Computability and recursion theory 2004h:03089 03D28 03B25 03E5S0 Montalban, Antonio (1-CRNL; Ithaca, NY Embedding jump upper semilattices into the Turing degrees. (English summary) J.  ...  Let Z be a jusl of Turing degrees. The author proves that every countable upper semilattice can be embedded in Z.  ... 

Density of a final segment of the truth-table degrees

Jean-Leah Mohrherr
1984 Pacific Journal of Mathematics  
We give a simple and explicit example of elementary inequivalence of the Turing degrees to the truth-table degrees.  ...  In constructing this example, we show that every truth-table degree above that of the halting problem is the jump of another truth-table degree.  ...  A partial ordering is an upper semilattice if finite supremums are always defined. An upper semilattice JSf is homogeneous if any embedding into <£?  ... 
doi:10.2140/pjm.1984.115.409 fatcat:h2pewm3ddvfcdivrhpb4djjpgm

Page 6761 of Mathematical Reviews Vol. , Issue 87m [page]

1987 Mathematical Reviews  
There is a natural embedding of the (total) Turing degrees into the partial degrees, and hence into the enumeration degrees (e-degrees), these being notational variants of the partial degrees as defined  ...  Finally, paralleling Sacks’s jump theorem for r.e. Turing de- grees, the author shows that the jumps of the I, degrees are ex- actly the Ilz degrees > 0’.  ... 

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

A note on continuities of the poset of Turing degrees

Luoshan Xu
2014 International Journal of Contemporary Mathematical Sciences  
In this note, continuities of the poset of Turing degrees are considered.  ...  ideal completion of D, and thus is an algebraic lattice; (4) D can be embedded into an algebraic lattice as an embedded base.  ...  Consequently, D can be embedded into the algebraic lattice as an embedded base.  ... 
doi:10.12988/ijcms.2014.4777 fatcat:7ot4pjkfn5fsnlq7wk6qudy2sa

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

1975 Mathematical Reviews  
The question on p. 309, line 12 has subsequently been answered by Lachlan and Lebeuf; every countable upper semilattice can be embedded as an initial segment of the degrees.  ...  In the present paper the author studies similar questions for the upper semilattices L(S) of classes of equivalent numerations for nontrivial sets S and the upper semilattices L.  ... 

Local Initial Segments of The Turing Degrees

Bjørn Kjos-Hanssen
2003 Bulletin of Symbolic Logic  
AbstractRecent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing  ...  degrees are indicated.  ...  This article concerns the algebraic study of the upper semilattice of Turing degrees.  ... 
doi:10.2178/bsl/1046288724 fatcat:rrfet36v4ngvrjsrkwv3r4i3le

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

1993 Mathematical Reviews  
First it is shown that the non-p-generic degrees are dense in R (the upper semilattice of r.e. degrees).  ...  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.  ... 


Stephen G. Simpson
2008 Computational Prospects of Infinity  
Motivation Recall that D T is the upper semilattice consisting of all Turing degrees.  ...  Note that R T , the upper semilattice of recursively enumerable Turing degrees, is embedded in P w . Moreover, 0 ′ and 0 are the top and bottom elements of both R T and P w .  ... 
doi:10.1142/9789812796554_0017 fatcat:p2ovjhh4lrefbabj4we22svqlq

JSL volume 68 issue 3 Cover and Front matter

2003 Journal of Symbolic Logic (JSL)  
MIRAGLIA : 946 A hierarchy for the plus cupping Turing degrees, by YONG WANG and ANGSHENG LI 972 Embedding jump upper semilattices into the Turing degrees, by ANTONIO MONTALBAN 989 Classical and constructive  ...  The BULLETIN and the JOURNAL are the official organs of the Association for Symbolic Logic, an international organization for furthering research in symbolic logic and the exchange of ideas among mathematicians  ... 
doi:10.1017/s0022481200008458 fatcat:s5wpgx2x5rarxiw52kt4h3iuq4

A Rigid Cone in the Truth-Table Degrees with Jump [chapter]

Bjørn Kjos-Hanssen
2016 Lecture Notes in Computer Science  
The automorphism group of the truth-table degrees with order and jump is fixed on the set of degrees above the fourth jump, 0 (4) .  ...  At separate stages of the preparation of this article the author was supported by a Marie Curie Fellowship of the European Community Programme "Improving Human Potential" under contract number HPMF-CT-  ...  The following lemma will have many applications: Lemma 5.7. Let U ⊆ ω. Acknowledgments We thank Noam Greenberg for a correction to an early draft of this article.  ... 
doi:10.1007/978-3-319-50062-1_29 fatcat:voiiugm2djdw7kiex4ohuw3rtm

A rigid cone in the truth-table degrees with jump [article]

Bjørn Kjos-Hanssen
2010 arXiv   pre-print
The automorphism group of the truth-table degrees with order and jump is fixed on the set of degrees above the fourth jump of 0.  ...  This material is based upon work supported by the National Science Foundation under Grants No. 0652669 and 0901020.  ...  It is not known whether the structure of the Turing degrees is rigid, but it is known [JS] that the structure of the Turing degrees with jump contains a rigid cone.  ... 
arXiv:0901.3949v3 fatcat:5i7u7i7pz5h5vh6fo5m4itodwy

Intervals Without Critical Triples [chapter]

Peter Cholak, Rod Downey, Richard Shore
1998 Logic Colloquium '95  
This paper is concerned with the construction of intervals of computably enumerable degrees in which the lattice M 5 (see Figure 1 ) cannot be embedded.  ...  Actually, w e construct intervals I of computably enumerable degrees without any w eak critical triples (this implies that M 5 cannot be embedded in I, see Section 2).  ...  Let a, b 0 and b 1 be elements in any upper semilattice L (such as the Turing degrees or the computably enumerable Turing degrees).  ... 
doi:10.1007/978-3-662-22108-2_2 dblp:conf/logicColl/CholakDS95 fatcat:bq2bw4kibbaxhitnoaekqludge

Intermediate logics and factors of the Medvedev lattice [article]

Andrea Sorbi, Sebastiaan A. Terwijn
2006 arXiv   pre-print
We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them.  ...  In our case it suffices to embed F ω as an initial upper semilattice of the Turing degrees. Notice that the range of such an embedding consists of Muchnik degrees.  ...  U embeds into Fr × (U) as an upper semilattice, and for every distributive lattice L, if f : U −→ L is a homomorphism of upper semilattices, then the embedding of U into Fr × (U) extends to a unique lattice  ... 
arXiv:math/0606494v1 fatcat:3spff4xw3vae5ikudiveba77qy
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