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Recursively enumerable sets and degrees

Robert I. Soare
<span title="1978-11-01">1978</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hjoli2j6qffdpaalkszryuidk4" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
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.  ...  Sacks has remarked that recursion theory is the heart of logic, and recursively enumerable sets form the soul of recursion theory.  ...  Hence B and B must be high. Stob [St] proved that B must be a major subset of some C C^ co. Define the relation = r on r-maximal sets by A = r B iff A n B is rmaximal.  ... 
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Page 2487 of Mathematical Reviews Vol. , Issue 81G [page]

<span title="">1981</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
We characterize the r.e. sets A with some BC,,,A as those with a A, function that for each recursive R, specifies R, or R, as infinite on A and to be preferred in the construction of B.  ...  Horst Reichel (Magdeburg) Lerman, Manuel; Shore, Richard A.; Soare, Robert I. r-maximal major subsets. Israel J.  ... 
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Determining automorphisms of the recursively enumerable sets

Richard A. Shore
<span title="1977-02-01">1977</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
Thus, for example, consider an r-maximal set A (no recursive  ...  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. Theorem.  ...  One of the major areas of concern in recursion theory has traditionally been the structure of recursively enumerable sets as a lattice, S, and, more particularly, that of S *, the lattice modulo finite  ... 
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Determining Automorphisms of the Recursively Enumerable Sets

Richard A. Shore
<span title="">1977</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
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.  ...  One of the major areas of concern in recursion theory has traditionally been the structure of recursively enumerable sets as a lattice, S, and, more particularly, that of S *, the lattice modulo finite  ...  Of course one can ask the same questions for other reasonable subclasses of set splits A) and a major subset B of A (A \J W =* N => B \j W = * N).  ... 
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A note on r-maximal subspaces of V∞

David R. Guichard
<span title="">1984</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
Main theorems Kalantari [3] defined the notion major subspace analogous to major subset in ordinary recursion theory, and showed that every r.e. non-recursive space has a major subspace.  ...  If I is a set, we denote by 1(s) or I" the subset of I constructed by stage s or enumerated after s steps in some fixed enumeration. For a space W, W" will usually mean (I-')* for some basis 1 of W.  ... 
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Page 605 of Mathematical Reviews Vol. , Issue 87b [page]

<span title="">1987</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
of the re- cursively enumerable sets connected with the major subsets (e.g., major subsets of hyperhypersimple sets or r-maximal sets).  ...  605 03D Recursion theory In the first section the author discusses in general the meaning of the major subsets inside the lattice of recursively enumerable sets, and the automorphism and isomorphism problems  ... 
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Page 5 of Mathematical Reviews Vol. , Issue 81K [page]

<span title="">1981</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
We study the possible values of the index set of »@ for a given family @.” {For the entire collection see MR 81a:68007.} 81k:03042 Chong, C. T. Major subsets of a-recursively enumerable sets.  ...  .,.) of maximal spaces, maximal spaces with no extendible bases, and even supermaximal spaces has been shown, and there is a very nice result of R. A. Shore [J.  ... 
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Page 818 of Mathematical Reviews Vol. , Issue 81C [page]

<span title="">1981</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The authors prove the following: There is an atomless hyperhypersimple set H, a splitting H,, H, of H, and an r.e. subset K of H such that K isa small r-maximal major subset of H, and for any r.e. set  ...  Lachlan’s proof used much of the structure theory of &, such as the existence of maximal sets and of small major subsets of r.e. nonrecursive sets. The authors require even more structure results.  ... 
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The intervals of the lattice of recursively enumerable sets determined by major subsets

Wolfgang Maass, Michael Stob
<span title="">1983</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
Theorem 1.1. Suppose B cm A. Then there is an r.e. set C such that B C_ CE A, C-B is infinite, and for every r.e. W, if WZ A -C, then WZ C-B.  ...  A first corollary of our theorem is that each r-maximal set B which arises in this way as a major subset of a maximal set has the same Z?*(B). For let M* = &?  ...  A has an r-maximal major subset B ifi there is a &ideal 9 of b;(A) which is maximal and which contains each recursive R such that R E" A. If B c rm A, then B is 'close' to A.  ... 
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Monotone and 1–1 sets

D. B. Madan, R. W. Robinson
<span title="">1982</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qucxg3yndjaptj6krclryhveua" style="color: black;">Journal of the Australian Mathematical Society</a> </i> &nbsp;
An infinite subset of u is monotone (1-1) if every recursive function is eventually monotone on it (eventually constant on it or eventually 1-1 on it).  ...  A recursively enumerable set is co-monotone (co-1-1) just if its complement is monotone (1-1).  ...  Recall that C is a major subset of D if C C D, D -C is infinite, and for every r.e. set W we have D' C *W implies C C *W. THEOREM 3.4. / / W is co-1-1 and S is a major subset of W then S is co-1-1.  ... 
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On Modal Logics of Partial Recursive Functions [article]

Pavel Naumov
<span title="2004-07-12">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established.  ...  The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations.  ...  The two major cases that will be considered are: a) {ξ u } u∈U is an enumeration of all nondeterministic partial recursive functions and b) {ξ u } u∈U is an enumeration of all deterministic partial recursive  ... 
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Page 485 of Mathematical Reviews Vol. 37, Issue 3 [page]

<span title="">1969</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(recursively enumerable) sets modulo the finite sets, and let A(R*) be the Boolean algebra generated by R*.  ...  (For example, Friedberg’s theorem, the one that asserts the existence of maximal r.e. sets, is an 3V-sentence.)  ... 
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Page 419 of Mathematical Reviews Vol. , Issue 81B [page]

<span title="">1981</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The notions of maximal and r-maximal sets are modified for subalgebras as follows. For subalgebras B and C write B=,C to mean that there are finite sets 7, and 7, such that (BU T7))*= (CU T,)*.  ...  For it is easily seen that a major subalgebra of a maximal subalgebra of ® must itself be an r-maximal subalge- bra of B. John W. Berry (Gaborone) Kechris, Alexander S.  ... 
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Page 1005 of Mathematical Reviews Vol. 38, Issue 6 [page]

<span title="">1969</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
set is high if and only if every deficiency subset is a major subset ; (iv) a maximal set can be either high or low.  ...  Every non- empty recursively enumerable set of non-negative integers is the range of some elementary definable function of one variable.  ... 
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Page 716 of Mathematical Reviews Vol. 56, Issue 3 [page]

<span title="">1978</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
M.; Nadyrov, R. F.; 5247 Solov'ev, V. D. A criterion for the completeness of recursively enumerable sets, and some of a fixed point theorem. (Russian) Izv. Vys§. Uéebn. Zaved.  ...  The main theorem proved states that there is no maximal | m-degree belonging to the btt-degree of a nonrecursive recursively enumerable set. The proof is elementary. Stephen L.  ... 
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