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Page 1634 of Mathematical Reviews Vol. , Issue 81E [page]

1981 Mathematical Reviews  
Symbolic Logic 39 (1974), 97-104; MR 50 #4266] showed that, if @(C) is a Boolean combination of r.e. sets, then C is a Boolean combination of open classes.  ...  Then 0,(C) is a Boolean combination of r.e. sets if and only if C is a Boolean combination of “small” open classes. Precise definitions may be found in the paper.  ... 

Page 12 of Mathematical Reviews Vol. 55, Issue 1 [page]

1978 Mathematical Reviews  
Moreover, in the cases in which m<n, the minimum length is realized by a Boolean combination of recursive open sets together with a single co-r.e. open set, where by definition the open set determined  ...  combination of recursive open sets representing &.  ... 

Page 591 of Mathematical Reviews Vol. 50, Issue 3 [page]

1975 Mathematical Reviews  
Conversely, it is shown that if % is in the (finite) ErSov hierarchy then ¢ is a Boolean combination of open sets; it is left open whether ¢ must be a Boolean combination of r.e. open sets.  ...  If ¢ is a Boolean combination of r.e. open sets, an upper bound is also obtained. In the case where the open sets are recursive, these bounds coincide, and an exact classification results.  ... 

Page 1738 of Mathematical Reviews Vol. , Issue 80E [page]

1980 Mathematical Reviews  
An effectively open set is the union of a recursively enumerable subset of the base; the effectively closed sets are the complements of the effectively open sets.  ...  Let 8 be such a Boolean algebra. Let L(%) denote the lattice of subalgebras of 8 having recursively enumerable basic sets.  ... 

Page 2295 of Mathematical Reviews Vol. , Issue 86f [page]

1986 Mathematical Reviews  
An r.e. set A has the universal splitting property (USP) if for any r.e. set D <7 A there is a splitting B and C of A (that is, BNC = © and BUC = A) such that B=r D.  ...  He then (Section 3) shows how this construc- tion can be easily combined with other requirements to yield such results as various strengthenings of the Ladner-Sasso result that every nonzero r.e. degree  ... 

Page 3652 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
Combining this with a dual the- orem by Ambos-Spies, Lachlan and Soare for r.e. degrees cupping to 0’, it follows that any open formula F(x, y) with two free vari- ables in the language of the r.e. degrees  ...  ordering of sets of any complexity type under inclusion (modulo fi- nite sets) is dense; and that the countable atomless Boolean algebra can be embedded in the structure of sets of the same complexity  ... 

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

Carl G. Jockusch, Jeanleah Mohrherr
1985 Proceedings of the American Mathematical Society  
to an equivalent finite Boolean combination of questions of the form "/c e BT' Then, A, B are said to have the same tt-degree if each is tt-reducible to the other, and tt-degrees have a natural ordering  ...  For sets A, B ç co, we say that A is a truth-table (tt) reducible to B if there exists an effective procedure for reducing any question of the form "m e A?"  ...  The results of this paper show that the two-atom Boolean algebra and various other finite lattices are in P. It is an open question whether all finite lattices are in P.  ... 
doi:10.1090/s0002-9939-1985-0781069-3 fatcat:qwodvdpzznbvdptmvzptsjax5m

Automorphisms of the lattice of recursively enumerable sets

Robert I. Soare
1974 Bulletin of the American Mathematical Society  
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.  ...  An r.e. set A is maximal if A* is a coatom (maximal element) of (A)=B (0(^4*)= 5*). A permutation p of N induces an automorphism O of S {$*) if AMS (MOS) subject classifications (1970).  ...  Combining this with results of Cooper and Jockusch we see that for a^O', a' = 0"o(3 infinite set A e 2L)\£ A is a Boolean algebra].  ... 
doi:10.1090/s0002-9904-1974-13350-1 fatcat:uyrxbaijizc5pektmahv3qkn6u

Page 2834 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
Each of the parts concludes with some open problems.  ...  An operator ® is 1-REA if it is an r.e. operator. ® is n-REA for n > 1 if, for all sets X, there is an index e such that 6(X) = X @W, where {W; } is a standard enumeration of the r.e. operators and X is  ... 

Page 6086 of Mathematical Reviews Vol. , Issue 87k [page]

1987 Mathematical Reviews  
Boolean prealgebra is universal with respect to the class of all r.e. preorders. Some re- finements of this result are examined.  ...  These subspaces of Voo are natural analogues of recursive subsets of w. The set of r.e. subspaces forms a lattice L(V.) and the set of decidable subspaces forms a lower semilattice S(V..).  ... 

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.  ...  The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10.  ...  For any coinfinite r.e. set A, A is hh-simple iff t(A) (or equivalentlySj) is a Boolean algebra.  ... 
doi:10.1090/s0002-9904-1978-14552-2 fatcat:arxp4btvhzfzjeoufybmnemt2u

Page 648 of Mathematical Reviews Vol. , Issue 88b [page]

1988 Mathematical Reviews  
Other applications include the lattice of r.e. subspaces of an effec- tively given infinite-dimensional vector space over a recursive field, the lattice of r.e. subalgebras of appropriately presented Boolean  ...  The relation between the simple open sets as defined by Kalantari and Leggett and the creative open sets is also discussed.  ... 

Computable Partial Solids and Voxels Sets [chapter]

André Lieutier
1999 Lecture Notes in Computer Science  
Since a central notion of discrete geometry, voxel sets, can be used to define computable partial solids, this approach throws a bridge between discrete geometry and solid modeling in R n .  ...  The model of computable partial solids has been recently introduced in order to address computational geometry and solid modeling issues within the Turing model of computation.  ...  Acknowledgements Original contents in this paper are the result of our common work with Abbas Edalat [6] .  ... 
doi:10.1007/3-540-49126-0_27 fatcat:7pyzi56ds5grvmj2mgjgodrypm

Page 272 of Mathematical Reviews Vol. 18, Issue 4 [page]

1957 Mathematical Reviews  
Soc. 50 (1944), 284-316; MR 6, 29] on recursively enumerable (r.e.) sets of positive rational integers (p.i.).  ...  It is shown that a hypersimple set is a r.e. set with an infinite complement which is less dense than the set of all p.i.; also that, given a hypersimple set Ho, then by successively taking away single  ... 

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

1987 Mathematical Reviews  
In the present paper the following results are proved: (1) If A is any Boolean combination of recursively enumerable sets and is effectively immune, then the Turing degree of A is equal to 0’; (2) every  ...  A set G of recursively enumerable (r.e.) degrees generates the r.e. degrees R if every r.e. degree is in the closure of G under finite joins and meets.  ... 
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