Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

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 lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

Nonisomorphism of lattices of recursively enumerable sets

1993 ◽  
Vol 58 (4) ◽  
pp. 1177-1188 ◽  
Author(s):  
John Todd Hammond
Keyword(s):  
Boolean Algebra ◽  
Isomorphism Class ◽  
Boolean Algebras ◽  
Turing Degree ◽  
Symmetric Difference ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Definition Of

Let ω be the set of natural numbers, let be the lattice of recursively enumerable subsets of ω, and let A be the lattice of subsets of ω which are recursively enumerable in A. If U, V ⊆ ω, put U =* V if the symmetric difference of U and V is finite.A natural and interesting question is then to discover what the relation is between the Turing degree of A and the isomorphism class of A. The first result of this form was by Lachlan, who proved [6] that there is a set A ⊆ ω such that A ≇ . He did this by finding a set A ⊆ ω and a set C ϵ A such that the structure ({W ϵ A∣W ⊇ C},∪,∩)/=* is a Boolean algebra and is not isomorphic to the structure ({W ϵ ∣W ⊇ D},∪,∩)/=* for any D ϵ . There is a nonrecursive ordinal which is recursive in the set A which he constructs, so his set A is not (see, for example, Shoenfield [11] for a definition of what it means for a set A ⊆ ω to be ). Feiner then improved this result substantially by proving [1] that for any B ⊆ ω, B′ ≇ B, where B′ is the Turing jump of B. To do this, he showed that for each X ⊆= ω there is a Boolean algebra which is but not and then applied a theorem of Lachlan [6] (definitions of and Boolean algebras will be given in §2). Feiner's result is of particular interest for the case B = ⊘, for it shows that the set A of Lachlan can actually be chosen to be arithmetical (in fact, ⊘′), answering a question that Lachlan posed in his paper. Little else has been known.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Rice Theorems for ∑n-1 Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 794-805 ◽  
Author(s):  
Nancy Johnson
Keyword(s):  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
The Difference

In [3] Hay proves generalizations of Rice's Theorem and the Rice-Shapiro Theorem for differences of recursively enumerable sets (d.r.e. sets). The original Rice Theorem [5, p. 3G4, Corollary B] says that the index set of a class C of r.e. sets is recursive if and only if C is empty or C contains all r.e. sets. The Rice-Shapiro Theorem conjectured by Rice [5] and proved independently by McNaughton, Shapiro, and Myhill [4] says that the index set of a class C of r.e. sets is r.e. if and only if C is empty or C consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets. Since a d.r.e. set is a difference of r.e. sets, a d.r.e. set has an index associated with it, namely, the pair of indices of the r.e. sets of which it is the difference.

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (3) ◽  
pp. 681-694
Author(s):  
Anne Leggett ◽  
Richard A. Shore
Keyword(s):  
Lattice Theory ◽  
Lattice Structure ◽  
Sufficient Conditions ◽  
Recursion Theory ◽  
Recursively Enumerable ◽  
Countable Ordinal ◽  
Congruence Relations ◽  
General Program ◽  
Simple 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 some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

2016 ◽  
Vol 73 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Juan A. Nido Valencia ◽  
Julio E. Solís Daun ◽  
Luis M. Villegas Silva
Keyword(s):  
Recursion Theory ◽  
Recursively Enumerable ◽  
Horn Formulas ◽  
Recursively Enumerable Sets

On the orbits of hyperhypersimple sets

1984 ◽  
Vol 49 (1) ◽  
pp. 51-62 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Boolean Algebra ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractThis paper contributes to the question of under which conditions recursively enumerable sets with isomorphic lattices of recursively enumerable-supersets are automorphic in the lattice of all recursively enumerable sets. We show that hyperhypersimple sets (i.e. sets where the recursively enumerable supersets form a Boolean algebra) are automorphic if there is a -definable isomorphism between their lattices of supersets. Lerman, Shore and Soare have shown that this is not true if one replaces by .

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (2) ◽  
pp. 419-426
Author(s):  
Manuel Lerman
Keyword(s):  
Lattice Theory ◽  
Regular Cardinal ◽  
Finite Sets ◽  
First Order ◽  
Recursively Enumerable ◽  
Regular Cardinals ◽  
Constructible Sets ◽  
Set Inclusion ◽  
Order Language ◽  
Recursively Enumerable Sets

Let α be an admissible ordinal, and let (α) denote the lattice of α-r.e. sets, ordered by set inclusion. An α-r.e. set A is α*-finite if it is α-finite and has ordertype less than α* (the Σ1 projectum of α). An a-r.e. set S is simple if (the complement of S) is not α*-finite, but all the α-r.e. subsets of are α*-finite. Fixing a first-order language ℒ suitable for lattice theory (see [2, §1] for such a language), and noting that the α*-finite sets are exactly those elements of (α), all of whose α-r.e. subsets have complements in (α) (see [4, p. 356]), we see that the class of simple α-r.e. sets is definable in ℒ over (α). In [4, §6, (Q22)], we asked whether an admissible ordinal α exists for which all simple α-r.e. sets have the same 1-type. We were particularly interested in this question for α = ℵ1L (L is Gödel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ1L is, of course, such an α), there are simple α-r.e. sets with different 1-types.The sentence exhibited which differentiates between simple α-r.e. sets is not the first one which comes to mind. Using α = ω for intuition, one would expect any of the sentences “S is a maximal α-r.e. set”, “S is an r-maximal α-r.e. set”, or “S is a hyperhypersimple α-r.e. set” to differentiate between simple α-r.e. sets. However, if α > ω is a regular cardinal of L, there are no maximal, r-maximal, or hyperhypersimple α-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (ω) points the way.

Hyperhypersimple supersets in admissible recursion theory

1983 ◽  
Vol 48 (1) ◽  
pp. 185-192
Author(s):  
C. T. Chong
Keyword(s):  
Recursion Theory ◽  
Order Type ◽  
Complete Characterization ◽  
Point Of View ◽  
Extensive Literature ◽  
Careful Study ◽  
Finite Sets ◽  
Natural Class ◽  
Recursively Enumerable Set ◽  
Simple Sets

Let α be an admissible ordinal. An α-recursively enumerable set H is hyper-hypersimple (hh-simple) if its lattice of α-r.e. supersets forms a Boolean algebra. In [3], Chong and Lerman characterized the class ℋ() of hh-simple -r.e. sets as precisely those -r.e. sets whose complements are unbounded and of order type less than . Perhaps a nice example of such a set is {σ∣σ is not for any n < ω}. It follows that all hh-simple sets in are nonhyperregular and therefore of degree 0′. That ℋ() is a natural class to study can be seen from the role played by its ω-counterpart in the study of decision problems and automorphisms of ℰ*(ω), the lattice of ω-r.e. sets modulo finite sets (Soare [13] gives an extensive literature on these topics). In α-recursion theory the existence of hh-simple sets is not an all pervasive phenomenon, and there is as yet no complete characterization of the admissible ordinal α for which ℋ(α) is nonempty. While this situation is admittedly unsatisfactory, we feel that the lattice ℰ*(α) of α-r.e. sets modulo α*-finite sets for which ℋ(α) ≠ ∅ deserves a careful study. Indeed armed with some understanding over the last few years of the general theory of admissible ordinals, it is tempting to focus one's attention on some specific ordinals whose characteristics admit a more detailed analysis of the fine structure of sets and degrees. From this point of view, and ℋ() are natural objects of study since the former is a typical example of a non-Σ2-projectible, Σ2-inadmissible ordinal, while the latter is important for the investigations of automorphisms over ℰ*().

scholarly journals Four games on Boolean algebras

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3389-3395
Author(s):  
Milos Kurilic ◽  
Boris Sobot
Keyword(s):  
Boolean Algebra ◽  
Winning Strategy ◽  
Zero Element ◽  
Boolean Algebras ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Other Hand

The games G2 and G3 are played on a complete Boolean algebra B in ?-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive pn < p and Black chooses an in ? {0,1}. White wins G2 iff lim inf pin,n = 0 and wins G3 iff W A?[?]? ? n?A pin,n = 0. It is shown that White has a winning strategy in the game G2 iff White has a winning strategy in the cut-and-choose game Gc&c introduced by Jech. Also, White has a winning strategy in the game G3 iff forcing by B produces a subset R of the tree <?2 containing either ??0 or ??1, for each ? ? <?2, and having unsupported intersection with each branch of the tree <?2 belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G3 played on B. It is shown that ? implies the existence of an algebra on which these games are undetermined.

Recursion theory on orderings. I. A model theoretic setting

1979 ◽  
Vol 44 (3) ◽  
pp. 383-402 ◽  
Author(s):  
G. Metakides ◽  
J.B. Remmel
Keyword(s):  
Algebraic Closure ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Formal Definition ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Partial Orderings ◽  
Definition Of ◽  
Algebraically Closed Fields

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a setS, denoted cl(S), is given in §1, however in vector spaces, cl(S) is just the subspace generated byS, in Boolean algebras, cl(S) is just the subalgebra generated byS, and in algebraically closed fields, cl(S) is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting (whose precise definition will be given in §1) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl(S) = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings.

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