The common state of numbers

A third method for constructing atomless r-maximal sets is given in He.ta2]. There the notion "e-state of a number x after s enumeration steps of (We)e0" is general-ized to the "common e-state of nitely many numbers x¹:::xn after s enumeration steps".

Letx1:::xn,e and s be numbers. With stW(x1:::xnes) we denote the "common e-state of the number x1:::xn at step s" what is dened by

stW(x1:::xnes)(j) = 1 if x1 ²Wjs ^{^}:::^{^}xn ²Wjs

= 0 if for one i xi ²= Wjs 1in j e:

12

(The commone-state of one number of course coincide with the usual e-state of this number.)

In the following construction in opposite to the maximal set construction we move not only one number, but nitely many numbers by reason of a greater common e-state (than other ones).

Theorem 2.6

There exists an atomless, r-maximal set A^.

Proof.

Let ; IN² be the set^f(nm)²IN² :n m n0^g and

;i =^f(im) : im^g,i0. Thus ; = _i^S

0;i

.

Further we assume that we have a computable, well-ordering of order type ! between all nite sequences of pairs (nm)²;. (Thus knowing that there is a nite sequence satisfying a computable condition we can nd the smallest sequence w.r.t. to this ordering having this property.)

We shall construct stepwise the set A together with a mapping T : ; ^7!IN.

Construction.

Step 0:

LetA0 = and T0 be a computable bijection between ; and IN.

Step s + 1 :

Look if

(⁹n)(⁹en)(⁹(k0m0):::(knmn)²;all di erent)

n < ki i = 01:::n^{^}stW(Ts((0n)):::Ts((nn))es) <`

stW(Ts((k0m0)):::Ts((knmn))es)]:

(2.9)

If yes choose the smallest n (let n0 be this) and the smallest n + 1-tuple for this n0. Let be this (k00m00):::(k0nm0n).

Now dene

Ts+1((in0)) = Ts((k0im0i)) : i = 01:::n0: Let` = max^fm_0i:i = 01:::n0^g+ 1.

Ts+1((k0im0i) =Ts((` + i` + i)) : i = 0:::n0

Ts+1((` + k` + k)) = Ts((` + n0 +k + 1` + n0+k + 1)) k 0 Ts+1((ij)) = Ts((ij)) for all other pairs

As+1 =As^fTs((in0))i = 0:::n0^g:

Result.

LetA be _s^S

0As. By the e ectivity of the construction, A is c.e.

1) For every pair (nm) ² ; limsTs((nm)) exists (proof by induction). Suppose Ts+1((00)) ⁶= Ts((00)). Then Ts+1((00)) ² W0s. But this can happen at most one time. Hence limsTs((00)) exists.

Suppose the limit exists for all pairs (nm) with m < i. Let s0 be such that for ss0

13

Ts((nm)) = Ts0((nm)) for all (nm) with m < i. If for an s > s0 Ts+1((ki)) ⁶= Ts((ki)) for a k with 0k i then the common i-state of the Ts+1((ki)), 0 ki is greater than that of the Ts((ki))'s. But this can happen only nitely often. Thus also limsTs((ki)) exists.

LetT be the limit function of (Ts)s0. Since allTs are bijections between ; and As,T is a bijection between ; and A. From this we get that A is coinnite.

2) For every: AT(;i) is c.e.

This follows easily from the facts: T0(;i) is c.e. and only nitely many elements placed while the construction on pairs from ;i are moved to pairs from ;j,j < i, since in (2.9) n < ki is required. (By the construction no number from a place in ;i can be moved to a pair from ;k with i < k).

Thus also AT(;0):::T(;i) is c.e. for every i0.

3) For everye

(⁹¹i)(We^\T(;i)⁶=⁾T(;) We):

(2.10)

(Proof by induction over e). Let e be a number and suppose for all j < e (2.10) holds where e is replaced by j. Then for an j < e:

(i) there is an kj such that Wj AT(;0):::T(;k^j) or (ii) T(;) Wj.

From this it follows that there is a numberm0 such that for allmm0 in case (i) for alln, m0 nm T((nm)) =²Wj and in case (ii) for all n T((nm))²Wj.

Lets0 be such thatT((`k)) = Ts0((`k)) for all k < m0.

Suppose (S ⁹¹i)(We^\T(;i)⁶=). Then we can nd ans s0 andm0+1 numbers from

i>m0;i such that their commonm0-state at steps is 1 for all j from (ii) or j = e. Thus this m⁰ + 1-tuple (or a smaller m⁰+ 1-tuple with the same m⁰-state is moved to the pairs (km0), 0k m0. This shows that T(;) We.

A is c.e. and coinnite by 1). A has no maximal superset. If A We ⁶= IN then by 3) there is an i s.th. We AT(;0):::T(;i). But then T(;i+1)^\We =. Hence We¹WeT(;i+1)¹ IN.

A is r-cohesive by 3). If R is a computable set with AR ⁶= IN then R AT(;⁰ :::;i) for somei. Hence A A R, by (2.10) (and thus A^\R = ). ² Later, e.g. in Lemma 3.3 we need a special property of the construction given in Theorem 2.6 (as also in Theorem 2.2). For i0 letAi be the set

fx : (⁹s)(x² Ts(;i)^{^}x²As+1)^g:

This means thatx comes to A at step s + 1 from a pair (im) for some m. Important are two properties of the sets Ai,i 0.

{ The sequence (Ai)i0 is an c.e. sequence and { Ri =AiT(;i) are c.e. sets for everyi0.

14

Remark 2.7

a) We see that if X is an innite subset of ; with (⁸i)(;i^\X ^; nite)

then T(X) is a cohesive set. This follows easily from property 3).

b) In Theorem 5.3 we shall modify the above construction in such a way that the nal functionT will be dened not for all pairs from ;. But the properties 1) to 3) true for all T((nm)), (nm)²; for which T((nm)) remains. Then we get

(^9<1i but not zero many)(T(;i) is innite)^{^} (⁹¹i)(T(;i) is not empty)⁾A²MSMax

(⁹¹i)(T(;i) is innite)⁾A²R^;Maxatl: This fact will be used there.

3 On the structures

^L(A)

for

^A

atomless,

^r

-maximal

One of the main topics, not only for the theory of the r-maximal sets, but for the general lattice analysis of the c.e. sets, is the description of the possibilities of the c.e.

superset structures of the atomless,r-maximal sets. Already from the basic construc-tions in subpoint 2 it can be concluded quite easily that there are more than only one isomorphism type of such lattices as we show. Let A and B be c.e. sets with A B and B^;A innite. A set C ^2L(AB) is called simple in^L(AB) if C ⁶= B and

(⁸D ^2L(AB))(D⁶= A ⁾D ^\C⁶= A):

(3.1)

(We see that a setS is simple in the usual meaning if S is simple in ^L(IN).)

The r-maximal set constructed in Theorem 2.4 is not dense simple, see Remark 2.5, 2). But every coinnite c.e. set A, not dense simple has in ^L(A) an element which is simple in ^L(A), see He.ta2]. On the other side the sets constructed in Theorem 2.2, as also in Theorem 2.6 do not have such special elements in their c.e. superset struc-tures. In Remark 3.2 the isomorphism type of these structures is given, from what the nonexistence of such relatively simple sets at once follows. ⁴⁾

Thus there are at least two isomorphism types of c.e. superset structures of atomless, r-maximal sets. In the following theorem we shall show that there are innitely many nonisomorphic such sublattices. But a complete description of all isomorphism types

L (A) for A²R^;Maxatl is still unknown.

4)In So87] it is remarked that by modifying the construction in Theorem 2.2 it can be constructed and atomless, r-maximal set A such that^L(A) has an simple set in ^L(A).

15

3.1 Basic calculation of

^L

^{( A )} , the number of nonisomorphic intervals

^L

^{( A )} 's, ^A

²

^R

^;

^Max

^atl

Probably the best method for characterizing the isomorphism types of ^L (A) for A²R^;Maxatl seems to be the form as it was done by Lachlan in La68a] for the c.e.

superset structures of the hh-simple sets. This was what we will call "Basic calcula-tion" of ^L (A). In this style the structures for the atomless, r-maximal sets A would be characterized by the following four conditions:

1 ^L (A) is an innite ⁹⁸⁹-lattice with 0 and 1, 2 ^L_r(A) consists of two elements (A and IN ),

3 (⁸B ^2L(A))(B ⁶= IN ⁾(⁹C^2L(A))(B ¹ C¹ IN)).

4 (⁸B ^{2 L}(A))(A ¹ B ¹ IN ^{) L} (AC) = ^Lms) (^Lms { the major subset interval).

We see easily that the conditions 1 to 4 are satised for every A ² R ^;Maxatl. The main still open question is if also the converse implication holds, i.e. if for every distributive lattice^Lwhich has the properties 1 to 4 there is anA²R^;Maxatl s.th.

L

=^L (A)?

*

Another possibility to classify ^L (A), A ² R^;Maxatl could be the "ideal characteri-zation". For A²R^;Maxatl let^L;(A) be

fB ^2L(A) : B ⁶= IN^g:

Obviously^L;(A) is a nonprincipal ideal (in ^L(A)).

LetX and Y be c.e. sets with X mY . Exists a characterization of nonprincipal ideals in ^L(XY ), e.g. in the form

fZ ^2L(XY ) : Z U^g

for setsU with X U Y such that the above ideals have the same isomorphism types as the ^L;(A)⁰s and U is from some arithmetical class?

*

A weaker fact was shown by Cholak and Nies in ChN.ta]. The following theorem improves the consideration in the introduction of this subpoint.

Theorem 3.1 (Cholak/Nies)

There are innitely many atomless, r-maximal sets with pairwise non-isomorphic c.e. superset structures.

16

Proof.

The construction of the atomless,r-maximal sets given here is a generalization of that from Theorem 2.2. We shall work with innite computable trees ; IN^<! and assume that an e ective coding ²;^$i ²IN for all notes from ;, except^hi, onto IN with the property

⁾i < i ²;^nfhig is given.

It will be constructred stepwise an 1^;1 function Ts : IN ^! IN, the set As (with As = rg(Ts)), both with a similar meaning as in Theorem 2.2, and a functionds :IN ^! IN with ds(m)m (more precisely ds(m) can have only at most two values for s0, m and a special number lesser than m). The meaning of ds is: While the construction we shall not try to maximalize them-state of Ts(m) resp. to st⁰ as in Theorem 2.2, but the ds(m)-state. Since for every n for a.a. m and a.a. s ds(m) > n, the constructed setA also will be (Un)n0-maximal (what is necessary for the r-maximality of A).

The setA will have the following properties:

(i) For ever ²;, ⁶=^hi the set A equal to A^fT(^hi`<sup>i</sup>) :`0 ^g will be c.e.

(ii) For ² ; and every j ² IN such that (j) ² ; A will be simple in

L(A^;A (j)).

(iii) The "tower sets" Hn for n 0 will be A^fA :in^g

(i.e. AWn Hn or AWn = IN for n 0).

(For ensuring the property (ii) we need the function dS. The sets A with i =n and Hn in general are di erent sets and have di erent meanings for A.)

We see that from (i) it follows for ²; :A^\A =A^\. We remember that every numberm is equal to a number of the form^hikⁱfor some unique numbersi and k and m is equal to a number of the form ^hx^hyzuⁱⁱ for some unique numbers x, y, z and u. We assume that both correspondences are bijective (the rst between IN and IN² and the second one betweenIN and IN⁴).

Construction

Step 0:

DeneT0(m) = m, m 0,A0 = and d0(m) = m, m0.

Step s ^{+ 1 :}

This step consists of two parts. First a) which givesTs⁺, A+s and d+s (by using Ts, As and ds) and then b) giving Ts+1, As+1 and ds+1 (by using Ts⁺, A_+s and d_+s).

17

Part a). Look if

(⁹e)(⁹i)(e < is^{^}st⁰(Ts(e)ds(e)s) < st⁰(Ts(i)ds(e)s)):

(3.2)

If yes let e0 be the smallest such e and for e0 let i0 be the smallest i s.th. (e0i0) satises (3.2). Now dene

A_+s = As^fTs(k) : e0 k s k ⁶=i0^g

Ts⁺(k) = Ts(k) : k < e0

Ts⁺(e0) = Ts(i0)

T_s⁺(e0+k) = Ts(s + k) : k > 0 d+s(m) = ds(m) for all m0:

If (3.2) does not hold let Ts⁺=Ts, A+s=As and d+s=ds. Part b). Look if

(⁹e)(⁹i)(e < is^{^} e =^hi^hjn`<sup>ii</sup> for some jn`^{^} i =^hi (j)kⁱ for some k^{^}

st⁰(Ts⁺(e)^hijnⁱs) st⁰(Ts⁺(i)^hijnⁱs)^{^} (⁸`<sup>0</sup>`)(Ts⁺(^hi^hjn`⁰ⁱⁱ)²=Wns^{^}Ts⁺(i)²Wns):

(3.3)

If yes nd the laste and i (rst e than i) s.th. (3.3) holds and dene As+1 = A+s^fTs⁺(k) : ek s k ⁶=i^g

Ts+1(k) = Ts⁺(k) : k < e Ts+1(e) = Ts⁺(i)

Ts+1(e + k) = Ts⁺(s + k) : k > 0 ds+1(e) = ^hijnⁱ

ds+1(m) = d+s(m) : (m)0 ⁶=i

ds+1(m) = m for all m = ^hi^hjn`<sup>00ii</sup>with `^{00 6}=`:

If (3.3) is not satised letTs+1=Ts⁺,As+1 =A+s and ds+1 =d+s.

Result.

Let A =_s^S

0As.

1. For every m limsTs(m) exists. (By induction over m.) We assume that this holds for all i < m. Let s0 be such that Ts+1(i) = Ts(i) for s s0 and i < m. If Ts+1(m)⁶=Ts(m) for ss0 happens two times by part b) then in the second case the

hijnⁱ-state of Ts(m) is greater than in the rst case. Since at least in the rst time after part b) in which part a) happens, dt(m) =^hijnⁱ. Thus, this can happen only nitely often. But case a) also can happen at most nitely often.

2. For all n for almost all m and almost all s ds(m) > n. By denition of ds+1 in part b) last line we see that among the numbers ^hi^hjn`ⁱⁱ, e 0 at most for one

18

ds+1(m) =^hijnⁱ and for all others ds+1(m) = m. But numbers^hxyzⁱ lesser than a given one are only nitely many.

3. A is (Un)n0-maximal.

In both construction parts we maximalize the ds(m)-state respectivley to (Un)n0 of Ts(m). By 2. we get the (Un)n0-maximality of A.

4. We have

(⁸n)(⁸i)(⁸j) Wn^\fT((^hi (j)`<sup>i</sup>) :` 0^g) is innite⁾ Wn^\fT((^hi`<sup>i</sup>) :` 0) is not empty]:

(3.4)

Assume not. Choose some n i and j such that (3.4) does not hold. Let be the

hijnⁱ-state of A, see 2. Then there is an m0 s.th. for m m0 all T(m) hve the

hijnⁱ-state .

Choose somem1 m0withm1 =^hi^hjn`ⁱⁱand somem2 > m1 withm2 =^hi (j)kⁱ and T(m2) ²Wn. Such numbersm1 and m2 exist by our assumption. Let s0 be such that for all i m2 Ts0(i) = T(i). Then for all s s0 (m1m2) satises (3.3), but never get attention, sinces s⁰ and the choice of s⁰. But this is not possible.

5. The setsAdened in (i) are c.e. sets. Given. Let be the i+1-state of A. Let m0 be s.th. for allm m0 T(m) has the i+1-state. Let s0 be s.th. T(k) = Ts0(k) fork m0 and part b) does not hold in a steps+1, ss0for a number^hijnⁱ < i

and a state ⁰.

Then A is equal (mod = ) to y : (⁹ts0)(y²At+1^_

(⁹k)(m0 < kt Tt(k) = y k =^hijⁱ for some ^{^} (⁸m)(m0< mk ⁾st⁰(Tt(m)it) = )^{^}

^(⁸mk)(Tt+1(m) = Tt(m)]:

6. The sets Hn =^SfA :i n^g form a tower withWn Hn or AWn= A.

Observe we have assumed ⁾i < i. Thus the sets Hn are c.e. The property (2.1) can be shown similar as in Theorem 2.2.

*

Let ;n = ^fn ² IN^<! : ^jj n^g. For all n 1 ;n are innite, computable trees.

Thus there are sets An constructed by means of ;n. We show that ^L (An)⁶= ^L (Am) for n⁶=m, nm1.

WithA_n, ²;n we denote the "basic sets" of An, n1, and for⁶=^hi withX_n the

"atom"An^;A_n^;. Suppose for n > m1 ! is an isomorphism between ^L (An) and

L (Am). Denote with , ²;n the set

f² ;m: !(X_n)^\X^m;1g: 19

1) is a nite, nonempty set.

SinceXn An and !(An) ^S

2

Am and is nite, see above, the set must be nite too.2) For nite, nonempty sets ¹² IN^<! we write ^{1 2} if

(^{8 2}2)(⁹v²1)(v):

For ²;n, ⁶=^hiand j ²;n it holds j:

Suppose not. Let ² j s.th. for all ^{0 ) 0 2}= . Take the set Am. This set is c.e., hence !^;1(A_m) also. We have !^;1(A_n)^\(A_{n j}^;A_n)^;1, since ² j. Thus !^;1(Am)^g\(An^;A_n^;)^;1. HenceAm^\(!(An)^;!(A_n^;)) is innite. This means that for some ⁰ X^{m0 \}!(X_n)^;1. Hence ⁰².

3) For every atom X^m only for nitely many²;n

X^m\!(Xn) is innite:

If not, since every An consists only of nitely many atoms (united with An) and X^m Am, !^;1(Am) would be not included into nitely many sets An. But this contradicts 6. in the theorem.

4) For every²;n, ⁶=^hi, ^jj< n there is an j with j ²;n s.th.

j

i.e. j^{^}(⁸v²)(^{8 2} j)(v ⁾v ). Take the set (this set is nite, by 1)) and consider ^fj : j^{\ 6}=^g. By 4), this set is nite. Thus even for almost all j we have

j ⁾ j:

But by 3) we have j. Hence j for almost all j.

5) For nite, not empty let ord() be equal to min^fjj:^2g:

We have for at least one with ² ;n, ⁶=^{h i} ord() ^jj. (By induction over the length of ² ;n). For ²;n with ^jj = 1 ord() 1, since Xn is an innite set and thus ⁶=. Having ord()^jj, by 4) we nd an j s.th. j. Hence ord( j)^jj+ 1.

Since for some ² ;n with ^jj = n ord() n, but m < n and thus ;m has no

element of order n, such mapping ! cannot exist. ²

Remark 3.2

The setA1 from Theorem 3.1 is of particulary interest. In ChN.ta] the atomless,r-maximal sets A having a sequence (Ci)i0 of c.e. supersets withCi^;A^;1 for everyi, Ci^\Cj =A for i⁶=j and

(⁸D ^2L(A))(D ⁶= IN ⁾(⁹n)(D C0C1:::Cn)) (3.5)

are called triangles. The set A1 is a triangle, since the sets A(j), j 0 have the properties mentioned above for (Ci)i0. We use^Tr as symbol for the class of triangles.

20

{ A sequence (Ci)i0 as above is called basis for ^L;(A). A basis for ^L;(B) with B triangle is not uniquely determined, but two basises are closely related. For D ^2L;(B) with D^;B^;1we say that D^02L(B) is a relatively splitting half of D if

(⁹D^{00 2L}(B))(D^0\D⁰⁰ =B^{^}D⁰D⁰⁰ =D):

If (Ci)i0 and (Ei)i0 are two basises for^L;(B) then every Ei is a nite union of splitting halfs of sets Ci and converse.

{ The class of triangles even is elementary denable in^E. It holds: "A is a triangle"

if A is an atomless, r-maximal set ^{^} (⁸V ^2L;(A))(⁹W ^2L;(A))V W ^{^} (3.6)

(⁸S^2L;(A))(⁹T ^2L;(A))(W ^\T = A^{^}S W T)]

) Let (Ci)i0 be a basis for ^L;(A). Given V ^2L;(A), by (3.5)

V C0 :::Cn for some n. Let W = C0 :::Cn. Hence V W.

If S ^{2 L;}(A) then S C0:::Cm for some m again by (3.5). Now let T = ^SfCi :n < i m^g if n < m and T = A otherwise. Then W ^\T = A by denition and S W T.

( By using (3.6) we can construct a basis for ^L;(A). Take some V ^2L;(A) with v^;A^;1. For this V exists W s.th. (3.6) is satised (for all S). Let C0 =W. Now take a V ^2L;(A) with V ^;C0^;1and chooseW for C0V . Then we deneC1 =W^;C0. Repeating this method innitely often, where every W ^2L;(A) is considered, we get a basis (Ci)i0 for ^L;(A).

By a similar condition as (3.6) also every set for ^L;(A) which is a member of a basis for ^L;(A) can be dened elementarily in^E.

{ From the denition of triangles it follows easily that if A is a triangle then

L (A)=^LW^Lms

where ^LW^Lms denotes the isomorphism type of the weak product of ^Lms { the major subset interval (mod = ).

Thus all triangles have isomorphic c.e. superset structures. Still open for triangles is their automorphism characterization, see the end of this subpoint.

That triangles do not have in their superset structures a set simple in them is obvious.

Since the atomless,r-maximal set constructed in Theorem 2.2 and also in Theorem 2.6 are triangles we showed here that what was mentionedin the introduction of subpoint 3.

*

21

After the triangles the next greater class of atomless,r-maximal sets which seems to be possible to characterize from the point of view of the possibilities of their c.e. superset structures is the class of major subsets of triangles.

Obviously every triangle is a major subset of a triangle, since the class ^Tr is closed upwards (inside the class of coinnite c.e. sets), but not converse.

LetA be a triangle and X smA. Then X is not a triangle. To see this take a coinnite c.e. supersetY of X. Then (Y ^\A)^;X is¹, what contradicts (3.6). For the elements from^MSTr (the major subsets of triangles) ^;Tr we do not have a characterization of the c.e. superset structures as for triangles (by one isomorphism type).

Lemma 3.3

There are elements X and Y from^MSTr^;Tr such that ^L(X)⁶=^L(Y ).

Proof.

LetA be the set from Theorem 2.6 which is a triangle, (Ai)i0 the c.e. sequence and (Ri)i0 the sequence dened there. (The sequence (ARi)i0is a basis for^L;(A).) Now let X sm A and any innite and coinnite c.e. set. Let Y be equal to

X ^fAi :i^2g:

Since (Ai)i0 is an c.e. sequence and is c.e.,Y also is c.e.

{ X =^2Tr, see above.

{ Y ^2MSTr^;Tr. X Y by denition. A^;Y ^;1, since Ai^;X^;1 and is coinnite. Thus Y mA, Y =^{2 T}r. Let B be coinnite with A B. If B would be a basis set for ^L;(Y ) then every other set C ^2L;(Y ) has the form

(C^\B)C⁰

withC^0\B = Y . But for an i with i =² and Ri^\B = A (such an i exists, since A^{2 T}r and ^;1) has the property Ri^\B ⁶= Y , since Ai^;X Ri^\B and (Ai^;x)^\Y =. Thus B is no basis set. But if Y would be from^Tr then every set from ^L;(Y ) is included into a basis set for ^L;(Y ).

{ ^L(X)⁶=^L(Y ). The set A is a set simple in ^L(X), but in^L(Y ) there are no such elements. If z ^{2 L;}(Y ) would be simple in ^L(Y ) then AZ much more would it be. But AZ AR0:::Rn for somen. Take i ² and i > n. Then

Ri^;Y ^;1, but Ri^\(AZ) = Y . ²

Interesting would be to have a complete description of all possible isomorphism types of ^L (X) for X ^2MSTr (mod^Lms), i.e. by using^Lms as parameter to characterize all

L (X)'s.

Question.

In HarN98] it is shown that in the major subset intervals "

*

₀₃-embedding is equal to computable embedding (mod = ), i.e. letAmB. Then

(⁸g T ⁰⁰)(⁹f rec.fct)(⁸n)(B^;A)^\Wg(n) = (B^;A)^\Wf(n)]:

Holds this property also inside^L(C) for C ²R^;Max?

22

A partial classication of R^;Max is of particularly interest. This is the description of all nonempty subclasses of R^;Max which can be dened by an ⁹⁸⁹-formula in the style as in La68b]. The basic structure is (IBA(^E )^E ), i.e. the Boolean closure of ^E with the unary relation ^E satised by the c.e. sets (mod = ) in the language for a Boolean algebra with a unary relation for ^E and all formulas have quantiers restricted to this predicate. What by using such formulas can dened in R^;Max? A rst view on this problem shows that there are more than ten di erent subclasses of R^;Max.

Im Dokument r-Maximal Sets (Seite 12-23)