A Note on Continuities of the Poset of Turing Degrees

reproduction in any medium, provided the original work is properly cited. Abstract In this note, continuities of the poset of Turing degrees are considered. Main results are: (1) The poset D of Turing degrees is an algebraic (and continuous) poset with a least element; (2) The poset D is not strongly continuous; (3) The directed completion of D is just the ideal completion of D, and thus is an algebraic lattice; (4) D can be embedded into an algebraic lattice as an embedded base.


Introduction
In mathematical logic, there is a branch of recursively (or computably) enumerable sets and degrees.For the poset D of Turing degrees and its subposet R of all r.e.degrees, there are many detailed studies on its logic aspects and order properties.The famous Sack's Density Theorem reflects the complexity of D in order aspects.The Lachlan-Lerman Theorem (see [4, shows that any countable atomless Boolean algebra B can be embedded into R, preserving sups, infs and least elements of R.
Different with discusses before, this note concerns continuities of the poset D and densely embedding D to other special lattices.We will see that D is continuous and algebraic but not strongly continuous.We also see that though D itself is not a lattice, D can be densely embedded into some algebraic lattices.

Preliminaries
We recall some notions and basic results.Most of them come from [2] and [3].
Let P = (P, ≤) be a poset.Then (P , ≥) is also a poset and called the opposed poset (or dual poset) of (P , ≤), denoted by P op .In a poset P , a nonempty subset D of P is directed if x, y ∈ D implies there exists z ∈ D with x ≤ z and y ≤ z.A nonempty subset F of P is filtered if it is directed in the dual poset P op .A dcpo means a poset in which any directed subset has a supremum.An ideal means a lower set which is directed.A filter of P means an ideal of P op .A principal ideal (filter ) is a set of the form ↓x = {y ∈ P : y ≤ x} (↑y = {x ∈ P : y ≤ x}).The notation sup x A denotes the supremum of A in the principal ideal ↓x.
Intuitively, we say that state x approximates state y if any computation of y yields the information of x at some finite stage.One of the important insights of the theory of "continuous posets" that has emerged in the last forty years is the following mathematical formalization.Definition 2.1.(see [3]) Let P be a poset, x, y ∈ P .We say that x approximates y, written x y, if whenever D is directed with sup D ≥ y, then x ≤ d for some d ∈ D. We use ↓ ↓x to denote the set {a ∈ P : a x}.If for every element x ∈ P , the set ↓ ↓x := {a ∈ P : a x} is directed and sup ↓ ↓x = x, then P is called a continuous poset.A continuous poset which is also a dcpo (resp., complete lattice) is called a continuous domain or briefly a domain (resp.continuous lattice).A poset P is said to be algebraic if every element of P is the directed supremum of compact elements.
A subset A of a poset P is Scott-closed if ↓A = A and for any directed set D ⊆ A, sup D ∈ A if sup D exists.All the Scott-closed sets of P in the order of set-inclusion is denoted by σ * (P ).It is easy to check that σ * (P ) is a complete lattice, called the Scott-closed set lattice of P .The complements of the Scott-closed sets form a topology, called the Scott topology, denoted by σ(P ).
In a topological space, a set A is said to be irreducible if for any pair of closed sets F 1 and F 2 with A ⊆ F 1 ∪F 2 , one always has that A ⊆ F 1 or A ⊆ F 2 .Definition 2.2.(see [6]) Let P be a poset and let c(P )(⊆ σ * (P )) be the set of all irreducible Scott-closed sets of P .Then in the set-inclusion order, c(P ) forms a dcpo, called the directed completion of P .
It is known (see [6]) that if P is a(n) continuous (resp: algebraic) poset, then c(P ) is a(n) continuous (resp: algebraic) dcpo.Definition 2.3.(see [8]) Let P be a poset and x, y ∈ P .We write x l y and say that x universally approximates y if, for any directed set D and any upper bound z of D such that y ≤ sup z D, there is d ∈ D such that x ≤ d.The subscript l means x approximates y locally and in the large.The set {y ∈ P |x l y} will be denoted ↑ ↑ l x and {y ∈ P |y l x} denoted ↓ ↓ l x.Definition 2.4.(see [8]) Let P be a poset.If for all x ∈ P , ↓ ↓ l x is directed and sup ↓ ↓ l x = x, then we say that P is a strongly continuous poset, or shortly, an SC-poset.Definition 2.5.(cf.[5]) The Scott topology on a poset P is called lower hereditary if, for every Scott-closed subset A, the relative Scott topology on A agrees with the Scott topology of the poset A.
Lemma 2.6.(see [5]) Let P be a poset.The following statements are equivalent: (1) P has a lower hereditary Scott topology; (2) For all x ∈ P , the inclusion map from the poset ↓ x into P is Scottcontinuous; (3) For z ∈ P and directed D ⊆ ↓z, x = sup z D implies x = sup P D.
It is known that every strongly continuous poset is a continuous one but not vice versa,.More precisely, we have Lemma 2.7.(see [8]) Let P be a poset.Then P is an SC-poset iff P is a continuous poset and has a lower hereditary Scott topology Definition 2.8.(see [2]) (a) A subset A of naturals is Turing reducible to the set It is known [2, Theorem 9-5.10] that, D is a sup-semilattice but a lattice.And it is know that a sup-semilattice which is also a dcpo must be not a complete lattice.By this observation, we can infer that the poset D of Turing degrees is not a dcpo.

Main Results
In this section, we will give our main results on the poset D of Turing degrees.
Firstly, it is know form Wikipedia Encyclopedia under the item "Turing degrees" that for each degree α, the set of degrees below α is at most countable, and there is no infinite, strictly increasing sequence of degrees that has a least upper bound.So, we have the following Proposition 3.1.(see [9]) In the poset D of Turing degrees, every ideal either has a largest element or has no least upper bound.
Proof.In terms of the above facts, it is a routing work by the contrary argument.
Theorem 3.2.In the poset D of Turing degrees, every element is compact and D is an algebraic poset.Especially, D is a continuous poset.Proof.By Proposition 3.1, if D in D is a directed subset with a supremum, than D has a largest element.Thus, every element in D is compact.This trivially follows that D is an algebraic poset.Definition 3.3.Let M be a proper ideal on a poset P .If there is a filter F s. t.M is maximal among the ideals which do not intersect F (for an ideal I on P , I ∩ F = ∅ and I ⊇ M imply I = M ), then we say M is a maximal ideal relative to filter F on poset P , or roughly, a maximal ideal relative to filters.Lemma 3.4.Let P be a poset, I an ideal and F a filter of P with I ∩ F = ∅.Then there always exists a maximal ideal M relative to filter F on P s.t.M ∩ F = ∅ and M ⊇ I.
Proof.Define A = {J : J be an ideal of P , J ∩ F = ∅ and J ⊇ I} .By the assumption, we have I ∈ A = ∅ and A is a poset in the set inclusion relation " ⊆ ".Let B be a linear subset of A. Let K = J∈β J. Claim that K ∈ A. It is clear that K is a lower set.For x, y ∈ K, there are J 1 , J 2 ∈ B s. t.
x ∈ J 1 , y ∈ J 2 .Since B is a linear subset of A, J 1 ⊆ J 2 or J 2 ⊆ J 1 .Suppose that J 1 ⊆ J 2 without losing generality.Then x ∈ J 2 and there is z ∈ J 2 ⊆ K s.t.x, y ≤ z.This shows that K is directed and K is an ideal of P .Since I ⊆ K and K ∩ F = ∅, K ∈ A. By the Zorn's Lemma, in A there is a maximal element M ∈ K.This M is indeed what we need a maximal ideal relative to filter F .Theorem 3.5.If α ∈ D is a limit degree, then there is an ideal I such that I ∩ ↑α = ∅ and sup α I = α.
Proof.Let P = (↑ α∪ ↓ α) ∩ D. Then P is a subposet of D. By Lemma 3.4, there is in P a maximal ideal I relative to the filter ↑α.It is easy to see that I is also an ideal of D, I ⊆↓ α and I ∩ ↑α = ∅.We need to show that sup α I = α.Firstly, α is an upper bound of I. Secondly, let degree β is another upper bound of I with β ≤ α.Suppose that β < α.Then it follows from the assumption that there is a degree γ such that β < γ < α.This contradicts to the maximality of I in P relative to the filter ↑α.So, β = α.This shows that sup α I = α.
It is known by the Sack's Density theorem [2, Theorem 9-5.15] that the first jump degree 0 is not a minimal cover of any degree which is strictly below it.This means that 0 is a limit r.e.degree.So, we have the following Theorem 3.6.The poset D is not an SC-poset.
Proof.For the limit degree α = 0 , using Theorem 3.5, we can obtain an ideal I satisfying that sup α I = α and α ∈ I.It follows from Proposition 3.1 that sup I does not exist.By Lemma 2.6(3), we see that the Scott topology on D is not lower hereditary.So, by Lemma 2.7, D is not strongly continuous, as desired.
Proposition 3.7.In a poset P with every element being compact, a set F ⊆ P is an irreducible Scott-closed set iff F is an ideal.Particularly, D is such kind of poset and the irreducible Scott-closed sets of D are exactly the ideals of D.
Proof.Since every element of P is compact, any ideal in P either has a largest element or has no least upper bound.So, the Scott-closed sets in P are exactly the lower set of P .By this observation, the if part is thus clearly true.To show the only if part, let F be an irreducible Scott-closed set in P .Then F is a lower set.What we need to show is that F is directed.To this end, suppose there are two elements x, y ∈ F with no common upper bound in F .Then F − ↑ x = F and F − ↑ y = F are Scott-closed sets and (F − ↑ x) ∪ (F − ↑ y) = F − (↑ x∩ ↑ y) = F , contradicting to the irreducibility of F .So, F is directed and is an ideal, as desired.
It follows from Theorem 3.2 that D is such kind of posets.
The ideal completion of a poset P is defined to be Idl(P ) = {I ⊆ P : I is an ideal in P } in the set-inclusion order.It is well-known that Idl(P ) is always an algebraic domain.Since D is a sup-semilattice with a lest element 0, it is easy to check that IdlD is an algebraic lattice.Definition 3.8.(See [3] for case of dcpos.)Let P be a poset, B ⊆ P .The set B is called a basis for P if ∀a ∈ P , there is a directed set D a ⊆ B such that ∀d ∈ D a , d P a and sup P D a = a, where the subscript P means to take relevant operations in the poset P .
It is well-known that a poset P is continuous iff it has a basis.To go further, we have Definition 3.9.(see [7]) Let B and P be posets.If there is a map j : B → P satisfying (1) j preserves existing directed sups, (2) j : B → j(B) is an order isomorphism, (3) j(B) is a basis for P , then (B, j) is called an embedded basis for P .If B ⊆ P and (B, i) is an embedded basis for P , where i is the inclusion map, then we say also that B is an embedded basis for P .
It is easy to see that if B ⊆ P , then B is an embedded basis for P iff B is a basis for P and for every directed set D ⊆ B with existing sup B D, one has sup B D = sup P D. If P has a basis B, then P is continuous.If B is an embedded basis for P , then B itself is also continuous.
Example 3.10.It is easy to see that the rationales Q is an embedded basis for the reals R. So, R and Q are continuous posets.Actually, it is easy to check that every linear ordered set is a continuous poset.Proposition 3.11.If P is a poset satisfying each element is compact, then (P, j) is an embedded basis for Idl(P ), where j : P → Idl(P ) is defined for all x ∈ P , j(x) =↓ x ∈ Idl(P ).
Proof.It is easy to check that j(P ) is a basis for Idl(P ).Since every element in P is compact, it is easy to check that j is Scott-continuous and j : P → j(P ) ⊆ Idl(P ) is an order isomorphism.So, by Definition 3.9, the proposition holds.
By Theorem 3.2 and Propositions 3. The above theorem has also an inference that if one requires a domain which has D as an embedded base, then the domain must be the algebraic lattice c(D) = Idl(D) up to an isomorphism.
(b) The sets A, B are Turing equivalent if A ≤ T B and B ≤ T A. We write this A ≡ T B. (c) Let A be a set of naturals.The equivalence class d T (A) = {B : B ≡ T A} is called the Turing degree of A. (d) The poset of all the Turing degrees with the partial ordering induced by the relation ≤ T is denoted by D, and letters α, β, γ, . . ., are used for Turing degrees.
7 and 3.11, we immediately have the following Theorem 3.12.The directed completion c(D) is just the ideal completion of Idl(D), i.e., c(D) = Idl(D).And c(D) is an algebraic lattice.Consequently, D can be embedded into the algebraic lattice as an embedded base.When the Scott topologies are concerned, the embedding D to its directed completion c(D) = Idl(D) is a densely embedding.