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Boolean algebras, splitting theorems, and Δ 2 0 sets
It is shown that every nonzero r.e. degree contains an r.e. set without the universal splitting property. That is, if δ \delta is any r.e. nonzero degree, there exist r.e. sets ∅ > T B > T A \emptyset > {}_TB > {}_TA with deg ( A ) = δ \deg (A) = \delta such that if A 0 ⊔ A 1 {A_0} \sqcup {A_1} is an r.e. splitting of A A , then A 0 ≢ T B {A_0}\not \equiv {}_TB . Some generalizations are discussed.
Let {We}e∈ω be a standard enumeration of the recursively enumerable (r.e.) subsets of ω = {0,1,2,…}. The lattice of recursively enumerable sets, , is the structure ({We}e∈ω,∪,∩). ≡ is a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V′ ∈ , if U ≡ U′ and V ≡ V′, then U ∪ V ≡ U′ ∪ V′ and U ∩ V ≡ U′ ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V]. We say that a congruence relation ≡ on is if {(i, j)| Wi ≡ Wj} is . Define =* by putting Wi, =* Wj if and only if (Wi − Wj)∪ (Wj − Wi) is finite. Then =* is a congruence relation. If D is any set, then we can define a congruence relation by putting Wi Wj if and only if Wi ∩ D =* Wj ∩D. By Hammond [2], a congruence relation ≡ ⊇ =* is if and only if ≡ is equal to for some set D.
The Friedberg splitting theorem [1] asserts that if A is any recursively enumerable set, then there exist disjoint recursively enumerable sets A0 and A1 such that A = A0∪ A1 and such that for any recursively enumerable set B
This paper analyzes the interrelationships between the (Turing) of r.e. bases and of r.e. splittings of r.e. vector spaces together with the relationship of the degrees of bases and the degrees of the vector spaces they generate. For an r.e. subspace V V of V ∞ {V_\infty } , we show that α \alpha is the degree of an r.e. basis of V V iff α \alpha is the degree of an r.e. summand of V V iff α \alpha is the degree and dependence degree of an r.e. summand of V V . This result naturally leads to explore several questions regarding the degree theoretic properties of pairs of summands and the ways in which bases may arise.
Let A be an infinite Δ₂⁰ set and let K be creative: we show
that K≤Q A if and only if K≤Q₁ A. (Here
≤Q denotes Q-reducibility, and ≤Q₁ is
the subreducibility of ≤Q obtained by requesting that
Q-reducibility be provided by a computable function f such that
Wf(x)∩ Wf(y)=∅, if x \not= y.) Using this result we prove
that A is hyperhyperimmune if and only if no Δ⁰₂ subset B of A
is s-complete, i.e., there is no Δ⁰₂ subset B of A such
that \overline{K}≤s B, where ≤s denotes
s-reducibility, and \overline{K} denotes the complement of K.
In [6], G. Metakides and the author introduced a general model theoretic setting in which to study the lattice of r.e. substructures of a large class of recursively presented models . Examples included , the natural numbers with equality, 〈 Q , ≤ 〉, the rationals under the usual ordering, and a large class of n -dimensional partial orderings. In this setting, we were able to generalize many of the constructions of classical recursion theory so that the constructions yield the classical results when we specialize to the case of and new results when we specialize to other models. Constructions to generalize Myhill's Theorem on creative sets [8], Friedberg's Theorem on the existence of maximal sets [3], Dekker's Theorem on the degrees of hypersimple sets [2], and Martin's Theorem on the degrees of maximal sets [5] were produced in [6]. In this paper, we give constructions to generalize the Morley-Soare Splitting Theorem [7] and Lachlan's characterization of hyperhypersimple sets [4] in §2, constructions to generalize Lachlan's theorems on the existence of major subsets and r-maximal sets contained in maximal sets [4] in §3, and constructions to generalize Robinson's construction of r -maximal sets that are not contained in any maximal sets [11] and second-order maximal sets [12] in §4.
In §1 of this paper, we give the precise definitions of our model theoretic setting and deal with other preliminaries. Also in §1, we define the notions of “uniformly nonrecursive”, “uniformly maximal”, etc. which are the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.
Let {We}e∈ω be a standard enumeration of the recursively enumerable (r. e.) subsets of ω = {0, 1, 2, …}. The lattice of recursively enumerable sets, is the structure ({We}e∈ω, ∪, ∩). is the sublattice of consisting of the recursive sets.
Suppose is a lattice of subsets of ω. ≡ is said to be a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V ∈ , if U ≡ U′ and V ≡ V′ then U ∪ U′ ≡ V ∪ V′ and U ∩ U′ ≡ V ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V].
The quotient lattices of (or of some sublattice ) correspond naturally with the congruence relations which give rise to them, and in turn the congruence relations of sublattices of can be characterized in part by their computational complexity. The aim of the present paper is to characterize congruence relations in some of the most important complexity classes.
We say that a pair of r.e. sets B and C split an r.e. set A if B ∩ C = ∅ and B ∪ C = A . Friedberg [F] was the first to study the degrees of splittings of r.e. sets. He showed that every nonrecursive r.e. set A has a splitting into nonrecursive sets. Generalizations and strengthenings of Friedberg's result were obtained by Sacks [Sa2], Owings [O], and Morley and Soare [MS].
The question which motivated both [LR] and this paper is the determination of possible degrees of splittings of A . It is easy to see that if B and C split A , then both B and C are Turing reducible to A (written B ≤ T A and C ≤ T A ). The Sacks splitting theorem [Sa2] is a result in this direction, as are results by Lachlan and Ladner on mitotic and nonmitotic sets. Call an r.e. set A mitotic if there is a splitting B and C of A such that both B and C have the same Turing degree as A ; A is nonmitotic otherwise. Lachlan [Lac] showed that nonmitotic sets exist, and Ladner [Lad1], [Lad2] carried out an exhaustive study of the degrees of mitotic sets.
The Sacks splitting theorem [Sa2] shows that if A is r.e. and nonrecursive, then there are r.e. sets B and C splitting A such that B < T A and C < T A . Since B is r.e. and nonrecursive, we can now split B and continue in this manner to produce infinitely many r.e. degrees below the degree of A which are degrees of sets forming part of a splitting of A . We say that an r.e. set A has the universal splitting property (USP) if for any r.e. set D ≤ T A , there is a splitting B and C of A such that B and D are Turing equivalent (written B ≡ T D ).
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 space V ∞ 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 set S , denoted cl( S ), is given in §1, however in vector spaces, cl( S ) is just the subspace generated by S , in Boolean algebras, cl( S ) is just the subalgebra generated by S , and in algebraically closed fields, cl( S ) is just the algebraically closed subfield generated by S .)
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 models which we study is that the algebraic closure of set is 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 of n -dimensional partial orderings.
Let V
∞ be a fixed, fully effective, infinite dimensional vector space. Let be the lattice consisting of the recursively enumerable (r.e.) subspaces of V
∞, under the operations of intersection and weak sum (see §1 for precise definitions). In this article we examine the algebraic properties of .Early research on recursively enumerable algebraic structures was done by Rabin [14], Frölich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8].In the main theorem below, we extend a result of Lachlan from the lattice of r.e. sets to . We define hyperhypersimple vector spaces, discuss some of their properties and show if A, B ∈ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if V ∈ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity.
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