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

1981
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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. ...##
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Page 12 of Mathematical Reviews Vol. 55, Issue 1
[page]

1978
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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 &. ...##
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Page 591 of Mathematical Reviews Vol. 50, Issue 3
[page]

1975
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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. ...##
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Page 1738 of Mathematical Reviews Vol. , Issue 80E
[page]

1980
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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*. ...##
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Page 2295 of Mathematical Reviews Vol. , Issue 86f
[page]

1986
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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 ...##
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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 ...

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

1985
*
Proceedings of the American Mathematical Society
*

to an equivalent finite

doi:10.1090/s0002-9939-1985-0781069-3
fatcat:qwodvdpzznbvdptmvzptsjax5m
*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. ...##
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Automorphisms of the lattice of recursively enumerable sets

1974
*
Bulletin of the American Mathematical Society
*

Let S denote the lattice

doi:10.1090/s0002-9904-1974-13350-1
fatcat:uyrxbaijizc5pektmahv3qkn6u
*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]. ...##
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Page 2834 of Mathematical Reviews Vol. , Issue 86g
[page]

1986
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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 ...##
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Page 6086 of Mathematical Reviews Vol. , Issue 87k
[page]

1987
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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..). ...

##
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Recursively enumerable sets and degrees

1978
*
Bulletin of the American Mathematical Society
*

The relation

doi:10.1090/s0002-9904-1978-14552-2
fatcat:arxp4btvhzfzjeoufybmnemt2u
*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. ...##
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Page 648 of Mathematical Reviews Vol. , Issue 88b
[page]

1988
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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. ...##
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Computable Partial Solids and Voxels Sets
[chapter]

1999
*
Lecture Notes in Computer Science
*

Since a central notion

doi:10.1007/3-540-49126-0_27
fatcat:7pyzi56ds5grvmj2mgjgodrypm
*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] . ...##
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Page 272 of Mathematical Reviews Vol. 18, Issue 4
[page]

1957
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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 ...##
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Page 4722 of Mathematical Reviews Vol. , Issue 87i
[page]

1987
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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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