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Page 768 of Mathematical Reviews Vol. 38, Issue 5 [page]

1969 Mathematical Reviews  
H. 4314 Degrees of recursively enumerable sets which have no maximal supersets. J. Symbolic Logic 33 (1968), 431-443. This impressive piece of work is a major extension of results of D. A. Martin [Z.  ...  set of degree a has a maximal superset.  ... 

Determining automorphisms of the recursively enumerable sets

Richard A. Shore
1977 Proceedings of the American Mathematical Society  
If G* is the class of recursive sets modulo finite sets or 911* C 6* C S * (91L* = maximal sets, S * = simple sets) then there is an automorphism of (the lattice generated by) G* which does not extend  ...  Upon first consideration it seemed that the answer to the first question should be no. Thus, for example, consider an r-maximal set A (no recursive  ...  A degree a contains a set A with exactly n maximal ior hhs) supersets iff a G L2 ithe class of low2 sets). Proof.  ... 
doi:10.1090/s0002-9939-1977-0446931-5 fatcat:o3q4gbdsoffqtpwulpn66pskwq

Determining Automorphisms of the Recursively Enumerable Sets

Richard A. Shore
1977 Proceedings of the American Mathematical Society  
We answer two questions of A. Nerode and give information about how the structure of S *, the lattice of r.e. sets modulo finite sets, is determined by various subclasses.  ...  A degree a contains a set A with exactly n maximal ior hhs) supersets iff a G L2 ithe class of low2 sets). Proof.  ...  Thus by Lemma 2 A has infinitely many maximal supersets. Conversely if a G L2 choose any set B (with recursive enumeration bis)) with exactly n maximal (hhs) supersets.  ... 
doi:10.2307/2041915 fatcat:cr3vvrjeqnazhbasyz4nfrom3e

Automorphisms of the lattice of recursively enumerable sets

Robert I. Soare
1974 Bulletin of the American Mathematical Society  
Maximal set* Post problem, high degree, low degree, lattice of r.e. sets. 1 The author owes much to the following who have supplied corrections, suggestions and information on this topic,  ...  Let S denote the lattice of recursively enumerable (r.e.) sets under inclusion, and let #* denote the quotient lattice of S modulo the ideal 3F of finite sets.  ...  For example, let M denote the class of degrees of maximal sets, and A the class of degrees of atomless sets, that is coinfinite r.e. sets which have no maximal supersets.  ... 
doi:10.1090/s0002-9904-1974-13350-1 fatcat:uyrxbaijizc5pektmahv3qkn6u

Recursively enumerable sets and degrees

Robert I. Soare
1978 Bulletin of the American Mathematical Society  
The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4.  ...  Sacks has remarked that recursion theory is the heart of logic, and recursively enumerable sets form the soul of recursion theory.  ...  Major subsets may be used to produce sets which fail to have r-maximal or AA-simple supersets.  ... 
doi:10.1090/s0002-9904-1978-14552-2 fatcat:arxp4btvhzfzjeoufybmnemt2u

Page 3145 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
Then he rewrites, with this new terminology, proofs of the following known results: the existence of a low simple set, the ex- istence of a DRE set which does not. have recursively enumerable degree, and  ...  The D-maximal sets are defined to be those r.e. sets A for which the cardinality of @(A) is two. Such sets are shhs, hence not diagonals.  ... 

Page 2912 of Mathematical Reviews Vol. , Issue 80H [page]

1980 Mathematical Reviews  
N. 80h:03062 m-degrees of supersets of simple sets. (Russian) Mat. Zametki 23 (1978), no. 6, 889-893.  ...  Lachlan showed that if B is a recursively enumerable (re) superset of a hyperhypersimple set A, then there exists a recursive set R such that RC B and B\ACR.  ... 

On the Lattice of Recursively Enumerable Sets

A. H. Lachlan
1968 Transactions of the American Mathematical Society  
This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper is to advance the search for a decision procedure for the elementary theory of r.e. sets.  ...  In §2 we prove the main theorem which states: if a is an r.e. subset of an r.e. set ß then either there exists a recursive subset 8 of ß such that a U S=ß or there exists a recursive sequence {S¡} of disjoint  ...  From Theorem 6 one can deduce exactly which complete extension of RD is the elementary theory of Jf* ; it turns out to be the one which is richest in structure.  ... 
doi:10.2307/1994768 fatcat:5flz4lxdhrabfaquddzcd4qcsa

On the lattice of recursively enumerable sets

A. H. Lachlan
1968 Transactions of the American Mathematical Society  
This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper is to advance the search for a decision procedure for the elementary theory of r.e. sets.  ...  In §2 we prove the main theorem which states: if a is an r.e. subset of an r.e. set ß then either there exists a recursive subset 8 of ß such that a U S=ß or there exists a recursive sequence {S¡} of disjoint  ...  From Theorem 6 one can deduce exactly which complete extension of RD is the elementary theory of Jf* ; it turns out to be the one which is richest in structure.  ... 
doi:10.1090/s0002-9947-1968-0227009-1 fatcat:bxaht6b5pfebxbxpwfmeeizrbu

Page 2487 of Mathematical Reviews Vol. , Issue 81G [page]

1981 Mathematical Reviews  
Recursively enumerable sets and degrees. Bull. Amer. Math. Soc. 84 (1978), no. 6, 1149-1181.  ...  hierarchy of classes of recursively enumerable degrees.  ... 

Automorphisms of the lattice of recursively enumerable sets: Orbits

R.G Downey, Michael Stob
1992 Advances in Mathematics  
Every non-low2 degree contains a set which is not haljhemimaximal. ProoJ Shoenfield [Sh] has shown that every nonlowz degree contains a set with no maximal superset.  ...  However, there are some r.e. degrees which contain no hemimaximal sets.  ... 
doi:10.1016/0001-8708(92)90065-s fatcat:53a7j4mdubgpnf5cidhlxp2p7m

Page 5795 of Mathematical Reviews Vol. , Issue 96j [page]

1996 Mathematical Reviews  
For example, every high computably enumerable degree contains a dense simple, strongly hsimple (these are “simplicity” properties) computably enumerable set such that no superset is hhsimple or r-maximal  ...  Turing reducibility with a recursive bound on oracle use). It is also shown that the truth-table degrees of non-low, computably enumerable sets are dense.  ... 

Page 5792 of Mathematical Reviews Vol. , Issue 91K [page]

1991 Mathematical Reviews  
The author proves that in every pair of r.e. nonrecursive Turing degrees there exist maximal r.e. equivalence relations which intersect trivially.  ...  The basic method for constructing a.n. sets and degrees is a modification of the ‘permit- ting method” of Yates which the authors call multiple permitting.  ... 

Page 485 of Mathematical Reviews Vol. 37, Issue 3 [page]

1969 Mathematical Reviews  
(A is hyper-hyper-simple in a superset B means that B-—A is infinite and there is no recursive sequence of dis- joint, finite subsets of B every one of which intersects B-—A.)  ...  (recursively enumerable) sets under inclu- sion.  ... 

Strong reducibilities

Piergiorgio Odifreddi
1981 Bulletin of the American Mathematical Society  
So we have a property (S is //-complete) which depends on the chosen enumeration of the r.e. sets.  ...  Note also that hyperhypersimple sets are such that every r.e. superset is complemented (they form a boolean algebra), whereas for r-maximal sets no nontrivial r.e. superset is complemented.  ... 
doi:10.1090/s0273-0979-1981-14863-1 fatcat:qbyl6mci6rd7jbxfwqjfmkit34
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