ON THE LATTICE OF RECURSIVELY ENUMERABLE SETS

@article{Lachlan1968ONTL,
  title={ON THE LATTICE OF RECURSIVELY ENUMERABLE SETS},
  author={Alistair H. Lachlan},
  journal={Transactions of the American Mathematical Society},
  year={1968},
  volume={130},
  pages={1-37}
}
  • A. Lachlan
  • Published 1968
  • Mathematics
  • 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. More precisely, an effective method is sought for deciding whether or not an arbitrary sentence formulated in the lower predicate calculus with sole relative symbol c is true of the r.e. sets. The main achievement of the paper is the characterisation of the hh-simple sets as those coinfinite r.e. sets… 

Nowhere simple sets and the lattice of recursively enumerable sets

  • R. Shore
  • Mathematics
    Journal of Symbolic Logic
  • 1978
Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the

An Overview of the Computably Enumerable Sets

  • R. Soare
  • Computer Science, Mathematics
    Handbook of Computability Theory
  • 1999

Types of simple α-recursively enumerable sets

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed

Automorphisms of the lattice of recursively enumerable sets. Part II: Low sets

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 A e ê let A* denote the equivalence

Degrees of recursively enumerable sets which have no maximal supersets

  • A. Lachlan
  • Mathematics
    Journal of Symbolic Logic
  • 1968
Two new theorems concerning the degrees of coinfinite recursively enumerable (r.e.) sets which have no maximal supersets are presented, showing that a″ = 0″ is sufficient for an r.e. degree a to be in A, and that a′ ≥ 0′ is necessary.

Recursion theory on orderings. II

  • J. Remmel
  • Mathematics
    Journal of Symbolic Logic
  • 1980
This paper gives the precise definitions of the model theoretic setting, constructions to generalize the Morley-Soare Splitting Theorem and Lachlan's characterization of hyperhypersimple sets, and describes the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

  • M. Lerman
  • Mathematics
    Journal of Symbolic Logic
  • 1971
The relationship between the elementary equivalence of f/.T and s/A, and the Turing degrees of A and B is studied, which shows that if a is the Turing degree of a maximal set, then there are infinitely many maximal sets M1, M2, .

Degree theoretic definitions of the low2 recursively enumerable sets

A major theme in recursion theory has been the investigation of the relation between a set's place in these orderings and other algorithmic, set-theoretic or definability type notions of complexity.

Lattices of α-Recursively Enumerable Sets

...

References

SHOWING 1-10 OF 13 REFERENCES

Three Theorems on Recursive Enumeration. I. Decomposition. II. Maximal Set. III. Enumeration Without Duplication

In this paper we shall prove three theorems about recursively enumerable sets. The first two answer questions posed by Myhill [1]. The three proofs are independent and can be presented in any order.

On the degrees of index sets. II

  • C. Yates
  • Mathematics, Computer Science
  • 1966
The main result of the present paper is the computation of the degree (in fact, isomorphism-type) of the index-set corresponding to the recursively enumerable sets of degree a: its degree is a(a); it follows from a theorem of Sacks that the degrees of such index-sets are exactly those which are > 0(3) and recursically enumerable in 0( 3).

A theorem on hyperhypersimple sets

The main concern of this paper is with demonstrating, and developing a few consequences of, what might be called the “density” of hyperhypersimple sets.

Recursively enumerable sets of positive integers and their decision problems

Introduction. Recent developments of symbolic logic have considerable importance for mathematics both with respect to its philosophy and practice. That mathematicians generally are oblivious to the

Review: Emil L. Post, Recursively Enumerable Sets of Positive Integers and Their Decision Problems

holds, e.g., of Ptolemy's principle that the earth is the center about which the heavenly bodies revolve. This principle could still be maintained if we were willing sufficiently to complicate the

Computability and Unsolvability

  • Martin D. Davis
  • Computer Science
    McGraw-Hill Series in Information Processing and Computers
  • 1958
Only for you today! Discover your favourite computability and unsolvability book right here by downloading and getting the soft file of the book. This is not your time to traditionally go to the book

Recursively Enumerable Sets and Retracing Functions

Three theorems on recursive enumeration

  • J. Symbolic Logic
  • 1958