r-Maximal major subsets
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂rm B) arose in attempts to extend Lachlan’s decision procedure for the αε-theory of ℰ*, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ*(A) and ℒ*(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂rm A as those with a Δ3 function that for each recursiveR i specifiesR i or \(\bar R_i \) as infinite on \(\bar A\) and to be preferred in the construction ofB. There are r.e.A andB with ℒ*(A) and ℒ*(B) isomorphic to the atomless Boolean algebra such thatA has anrm subset andB does not. Thus 〈ℰ*,A〉 and 〈ℰ*,B〉 are not even elementarily equivalent. In every non-zero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies \(H_1 \subseteq F \subseteq \bar L_2 \) .
- A. H. Lachlan,On the lattice of recursively enumerable sets, Trans. Amer. Math. Soc.130 (1968), 1–37. MR 37#2594. CrossRef
- A. H. Lachlan,The elementary theory of recursively enumerable sets, Duke Math. J.35 (1968), 123–146. MR 37#2593. CrossRef
- A. H. Lachlan,Degrees of recursively enumerable sets which have no maximal superset, J. Symbolic Logic33 (1968), 431–443. MR 38#4314. CrossRef
- A. H. Lachlan,On some games which are relevant to the theory of recursively enumerable sets, Ann. of Math. (2)91 (1970), 291–310. MR 44#1652. CrossRef
- M. Lerman,Some theorems on r-maximal sets and major subsets of recursively enumerable sets, J. Symbolic Logic36 (1971), 193–215. MR 45#3200. CrossRef
- M. Lerman, R. A. Shore and R. I. Soare,r-Maximal major subsets (Preliminary Report), Notices Amer. Math. Soc.23 (1976), A-650.
- M. Lerman and R. I. Soare,d-Simple sets, small sets and degree classes, to appear.
- M. Lerman and R. I. Soare,A decision procedure for fragments of the elementary theory of the lattice of recursively enumerable sets, to appear.
- D. A. Martin,Classes of recursively enumerable sets and degrees of unsolvability, Z. Math. Logik Grundlagen Math.12 (1966), 295–310. MR 37#68. CrossRef
- E. L. Post,Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc.50 (1944), 284–316. MR 6, 29. CrossRef
- R. W. Robinson,A dichotomy of the recursively enumerable sets, Z. Math. Logik Grundlagen Math.14 (1968), 339–356. CrossRef
- H. Rogers, Jr.,Theory of Recursive Functions and Effective Computability, McCraw-Hill, New York, 1967. MR 37#61.
- G. Sacks,Recursive enumerability and the jump operator, Trans. Amer. Math. Soc.108 (1963), 223–239. MR 27#5681. CrossRef
- J. R. Shoenfield,Degrees of Unsolvability, North-Holland, Amsterdam, 1971.
- J. R. Shoenfield,Degrees of classes of r.e. sets, J. Symbolic Logic41 (1976), 695–696. CrossRef
- R. A. Shore,Nowhere simple sets and the lattice of recursively enumerable sets, to appear in J. Symbolic Logic.
- R. I. Soare,Automorphisms of the lattice of recursively enumerable sets. Part I: Maximal sets, Ann. of Math.100 (1974), 80–120. CrossRef
- R. I. Soare,Automorphisms of the lattice of recursively enumerable sets: Part II: Low sets, to appear.
- M. Stob, Ph.D. Dissertation, University of Chicago, in preparation.
- r-Maximal major subsets
Israel Journal of Mathematics
Volume 31, Issue 1 , pp 1-18
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors