Saharon Shelah- If There is an Exactly λ-Free Abelian Group Then There Is An Exactly λ-Seperable...

8/3/2019 Saharon Shelah- If There is an Exactly -Free Abelian Group Then There Is An Exactly -Seperable One in

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IF THERE IS AN EXACTLY -FREEABELIAN GROUP THEN THERE IS AN

EXACTLY -SEPARABLE ONE IN

SH521

Saharon Shelah

Institute of MathematicsThe Hebrew University

Jerusalem, Israel

Rutgers UniversityDepartment of Mathematics

New Brunswick, NJ USA

Abstract. We give a solution stated in the title to problem 3 of part 1 of theproblems listed in the book of Eklof and Mekler [EM],(p.453). There, in pp. 241-242,this is discussed and proved in some cases. The existence of strongly -free ones was

proved earlier by the criteria in [Sh:161] in [MkSh:251]. We can apply a similar proofto a large class of other varieties in particular to the variety of (non-commutative)groups.

I thank Alice Leonhardt for the beautiful typingTyped December 12/93 - Done Fall 93;2 done 28 (+1) Feb.94last revision 3/13/95

Typeset by AMS-TEX

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2 SAHARON SHELAH

0 Introduction

Convention. In 0 and 1, group here means abelian group, and free meansin this variety.

We assume there is a -free, non-free (abelian) group of cardinality . We shallprove that there is a -separable non-free abelian group of cardinality , aprioria stronger statement. We rely on the characterization of as in the hypothesisfrom [Sh 161]: the existence of S, < s : < n >: Sf, (k) : k as there(see appendix; i.e. 3 here). Mekler Shelah [MkSh 251] dealt with a similar weakerproblem in a parallel way: if there is a -free not free abelian group of cardinality then there is a strongly -free one. In Eklof Mekler [EM], the present problem wasraised, discussed and sufficient conditions were given, depending on the form of S,see [EM], p.242-242, the problem in [EM],p.453. The direct sufficient condition is

that for every S Sf of cardinality there is a well ordering


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-FREE, -SEPARABLE 3

To accomplish () we need good control over how e.g. s ((0) > ) intersectB, and this is the information we put in the appendix on the -system (reallyold [Sh 161] is O.K., but we retain the appendix to ease reading).

0.1 Definition. For a set of variables, set of equations in some variables (maybeoutside ) let G(, ) be the (abelian) group freely generated by , except theequations in , i.e. the equations from mentioning only variables from .

0.2 Observation. 1) A sufficient condition (assuming sets of variables) for

() G(, ) is a free extension ofG(, ) (i.e. the mapping induced by id fromG(, ) into G(, ), which is always homomorphism, is an embedding, andG(, ) divided by the range of this mapping is a free group),

is

() there is an increasing continuous sequence : , 0 = , = ,and G(+1, ) is a free extension of G( , ).

2) Another sufficient condition for () of 0.2, is that by change of the variables in\, the set of equations is only .


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4 SAHARON SHELAH

1 Proving -Separability

Here we prove the main theorem; the reader is advised to look at 3.6, 3.7 at leastduring reading the beginning of the proof, and also to look again at the explanationin 0 of the proof when arriving to read the middle of the proof.

1.1 Definition. A group G is -separable if:H G,Rk(H) < H included in a free direct summand of G.(Remember: for an uncountable group H, its rank, Rk(H) is equal to its cardinality,|H|.)

1.2 Main Theorem. If there is a -free non +-free (abelian) group ( > 0,

necessarily regular) then there is a -free, -separable, not +

-free group.

Proof. The hypothesis of the theorem on the existence of such groups is analyzed indetail in [Sh 161] (most relevant are [Sh 161],3.6,3.7), and in particular, it impliesthe existence of n,S,(, S), B : Sc, s : Sf, < n with the propertiesas in [Sh 161],3.6,3.7 presented in 3.6, 3.7 of the Appendix here, and leta,m : m < list s

in increasing order for the order of B()() (see clause

(i) of 3.7) and without loss of generality we have in addition

() for Sf, < n, we have

,,m =: min{ : a,m B()} < ( , S)is non-decreasing in m,

and we call its limit (, ) (so s B()(,) and (, ) ()).

() if Si, , k = g() and cf((k)) = (, S) then

(a) for < (, S) we have sup{(, k) : Sf} is < (k)(b) the sequence min{(, k) : Sf} : < (, S) is strictly

increasing with limit (k).

(see Appendix, clauses (f)(), (f)() and (g) of 3.6).Let

K = {I :I Sc and [ = & I & I ( )] and

[ < > I & < < > I]},

K+ = {I K : I = and < > I / I]}.

For I K let J[I] =: { Si: for some , < > I}, so for J[I] there is aunique I[] (, S) such that [ < > I < I[]], note:I K+, J[I] I[] < (, S). For I K letSf[I] =: { Sf : for some k, k I} ; note that the k is unique and if


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I = {}, then k > 0, so we choose to write k = kI() + 1 (so forI = {}, kI() = 1). Also let

Y[I] =: {B : for some I we have Sc}

For Sf let wI() = { < n : for every m < , a,m Y[I]} (equivalently: for

infinitely many m < , a,m Y[I]).For every I K we define a group GI, it is freely generated byI =: {y,m : m < and Sf[I]} {x[a] : a Y[I]} except the equations (wecall this set I):

()I for Sf[I] and m < , the equation mI, defined as:

()mI, 2y,m+1 = y,m +

{x[a,m] : < n and a

,m Y[I]}

Note that 1 K, and let G = G(1); this abelian group is the example as in[Sh 161], Lemma 5.3, in particular G is not free. Let


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C Definition. 1) I1 I2 (from K) if Sf[I1] Sf[I2] and ( Sf(I1))[kI1() kI2()]. This implies Y[I1] Y[I2] and there is a clear relation between I1 andI2 : each equation =

mI1,

in I1 appears in I2 as = mI2,

but is

with more x[a,m]s (for same old but new s which appear because of some Sf[I2]\Sf[I1]) and I2 has members (not related to any equation from I1)involving a new . Another way to state this relation is( I1)( I2)[ ].2) I1 d I2 if I1 I2 and J[I1] is a


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J Definition. For I1 I2 (in K), satisfying I1,I2 below, let gI2,I1 be the homo-morphism from GI2 into GI1 defined by:

(i) ifa Y[I1] then gI2,I1(x[a]) = x[a](ii) ifa Y[I2]\Y[I1] then gI2,I1(x[a]) = 0(iii) if Sf[I1] then gI2,I1(y,m) = y,m(iv) if Sf[I2]\Sf[I1] and {a,k : k [m, ) and kI2 [] (equivalently

a,k Y[I2])} is disjoint to Y[I1] then gI2,I1(y,m) = 0

(this is enough for defining gI2,I1)where

I1,I2 for Sf[I2]\Sf[I1],

[kI2 (),n)

s is almost disjoint to Y[I1]

(i.e. has finite intersection).

K Fact. Assume I1 I2 are in K. Then

() gI2,I1 really defines a homomorphism which is onto (when I1 I2 and I1,I2holds)

() Kernel(gI2,I1) is the subgroup of GI2 generated by the set of x[a]s andy,m s which by Definition J are sent by gI2,I1 to 0.

Proof. Check the equations.

Main Fact L. If < and < > / S then G, is a direct summand ofG = G{}.

Proof. We can define by induction on k a number k n : k = 0, if k is definedand < n, let k+1 be the unique such that k < n and Sf cf((k)) =( , S) (exists by 3.3(f), all Sf behave the same by 3.6(a) (and see 3.2(6)(d)),note: if Sf cf((k)) = 0 then k+1 = n. Clearly if k is defined and < nthen k < k+1 n. So for some k

, k = n.We shall define by induction on k k the following Jk and, when k < k,

: Jk such that:

(0) Jk Sk

(1) < (, S) and / S for Jk(2) Jk+1 = { : S

k+1 and k Jk but (k) > k}(3) if Jk+1, k + 1 < k(), [, (, S)) and < > Sf then

sk Bkk is finite

(4) J0 = {}, = .

For k = 0 use clause (4). For k+1 we define Jk+1 by clause (2), now ifk+1 < k()for Jk

+1we have to find to satisfy clauses (1), (3), this is possible by (),()

in the beginning of the proof of Theorem 1.2.Let I0 = {< >: < },

I1 = { < >: for some k < k we have Jk and < },

I2 = I1\{< >: < },


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I3 = {< >: < = }.Note that by the inductive choice of the Jks:

if

S

f\S

f[I1

] then{

a,m

: < n and m < }

has a finite intersectionwith Y[I1].

(Use (3) noting that if Sf\Sf[I1] then (k) > k for every k < k such that

k Jk).Note also that: I0 K, I1 K+, I2 K+, I3 K+. Also I3 d I1 and I3 d I0and I2 I1 I0 (see Definition C(1)) and GI0 = G.

Note that gI0,I1 is well defined (see Definition J and Fact K).[Why? We have to check I1,I0 as defined there, but above says this]. Note alsothat gI1,I2 is well defined (again we have to check I2,I1 as defined in Definition

J, but for Sf(I1)\Sf(I2) by their definitions, (0) < so easily 0 (0) W0}

(B) assuming that the set of member of G is then{ < : is the set of elements of G,} is a club of .

Together with Main Fact L and Fact I, we are done. 1.3

1.4 Discussion. We can rephrase the proof of 1.1 combinatorially; i.e. explicitlywrite a set of generators X such that G = G XG, do not think it is clearer.To some extent this is done in Fact A of the proof of 2.2.


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2 The General Case: for a variety

We note here that a parallel theorem holds for any suitable variety consideringtwo variants of -separable (see Definition 2.1(2) and Definition 2.4). We do thegeneral case in less details.

2.1 Definition. 1) T is a variety if T is a theory (in a vocabulary ) all whoseaxioms are equations or just has the form x1, . . . , xn, an atomic formula.Without loss of generality every member of (function symbol or predicate) appearsin some axiom of T.2) A model M of T is called -separable if for every A M, |A| < we canrepresent M as a free product M1 M2 such that A M1 and M2 is free.

3) T has the n-th h-construction principle if we can find N, b,m(for < n, m < ) and Nm (for m n) such that:

(i) N a model of T of cardinality |T| + 0(ii) N is free, moreover, for each < n and m < we can complete

{b,n : < n, m < and [ = m < m]} to a free basis of N, call theset of additional elements C,m

(iii)() if mi = mi : < n n (for i = 1, 2) and m1 m2 (i.e.

( < n)(m1 m2 ) then Nm2 Nm1 N,

() b,m Nmk :k


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2.2 Theorem. Assume there is a -free not +-free abelian group exemplified byn, S, s : < n and Sf as in the proof of 1.2 and the theory T has the n-thh-construction principle and |T| < .

Then T has a -separable model of cardinality which is not free.

2.2A Conclusion. If there is a -free not +-free abelian group then for the varietyof groups (not the abelian one) there is a -free, -separable group G of cardinality which is not free. (I.e. G is a non-free group of cardinality , G can be represented

as


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(e) there is a list : < of S1\S0 without repetitions and () : <

such that 0 () < n and m() : < , m() < such that:

() {a(),m : m [m(), )} is disjoint to

Y0 {a,m : < n, < , m < }

() {a(),m : m < } Y1.

Proof. Included in the proof of Theorem 1.2.

Remark B. We can add

(f) S1 is Sf[I1] from the proof of Theorem 1.2 so for some function k from

S1\S0 to n = {0, . . . , n 1} we haveY1 = Y0 {a ,m : S1, m < and [k(), n)}.

Fact C. Under the conclusion of Fact A, lettingM0 =: {x[a] : a Y0} {y,c : S0, c N}M we have: M0 is free and forsome M2, M = M0 M2.

Proof. Clearly M1 is free (for T) as in the proof of Fact A in the proof of 1.2. Thenew point is to find M2.For each < n, m < , let C,m N be such that

C,m {b1,m1 : 1 = , m1 < or 1 = , m1 < m} is a free basis of N with norepetitions.

We let M2 be the submodel of M generated by:

(A) y ,c if < and c C(),m()

(B) x[a ,m] if < , a,m Y1\Y0 and for no <

do we have

a,m {a(k),k

: k [m(), )}(C) x[a] if a Y2\Y1(D) y,c if S2\S1, c Nm():


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Now < implies a / {a(),k

: k [m(), w)}.[Why? If < this is assumed in the case, if = this is follows by = (), andif (, ) this follows by clause (e)() (with s here standing for , there).

Hence the assumption of clause (B) holds.]By clause (B), x[a,m] M2 M0, M2.

Case 4: a ,m Y2\Y1.

By clause (C), x[a,m] M2 M0, M2.

Case 5: No previous cases.By the earlier cases = () and

{x[a1,m1 ] :

1 < n, m

1 < and [

1 = () m

1 < m()]} M0, M2.

Let N =: {c N : y ,c M0, M2}, so by the previous sentence

{b1,m1 : 1 < n, m1 < and 1 = () m1 < m()} N

, and by clause (A)also C(),m() N

hence (see clause (ii) in Definition 2.1) clearly N = N,

so x[a1,m] M0, M2 and y,c M0, M2.

We have proved {x[a] : a Y1\Y0} {x[a,m] : < n, m < , < }

M0, M2. As {x[a] : a Y0} M0 M0, M2 and by clause (C) we have{x[a] : a Y2\Y1} M0, M2 we conclude {x[a] : a Y2} M0, M2.

Also we have proved {y,c : c N, < } M0, M2 (this was done during

the proof of case 5) so {y,c : S1\S0 and c N} M0, M2.Also for S2\S1, letting N = {c N : y,c M0, M2},m = min{m : a

,m / Y1} we have: by clause (D), Nm:


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2.4 Definition. For a variety T and a model M of T and cardinality we say Mis -proj-separable, if for every A M, |A| < there is a free M M includingA and a projection h from M onto M.

Proof of 2.3. We define M, x[a], y,c as in the proof of 2.2. For every () < n andm() < , m() > 0 there is a homomorphism g(),m() from N onto b,n : 0. Let { : [

, )} list S2\S1.For each S2\S1 we can choose

() = k1, m() = Min{m : 0 < m < and a(),m / Y1}.

Let M1 be the model of T generated by1 = {x[a

,m] : < n, S1, m < } {y,c : c N, S2} freely except

1 =

y,c = x[a] : c = b,m, a = a

,m and S1

(y,c1, . . . , y,ck) : N |= (c1, . . . , ck), a T-atomic formula

.

Let M2 be the model of T generated by (note: I1, Jk are from the proof of 1.2)

2 =:

x[a] :a Y2 but if a B(,S),

g() = k1 and Jk1 J[I1] then a is in the first level

(i.e. like ak,0 ) or a B

( from the choice of I1)

freely except the equations

2 = 1 =

y,c = x[a] : c = b,m, a = a

,m , S1, < n, m < and x[a]

2

(y,c1 , . . . , y,ck) : N |= [c1, . . . , ck], a T-atomic formula, S1

.

(Note that if Jk1 and Sf then cf((k1)) = 0).Clearly M0 M1 M

2 M.

We define a homomorphism h2 from M into M2 : h2 M

2 is the identity, and for

= S2\S1 and c N we let:


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h2(y,c) = y,g(),m()(c).

Note: h2(x[a ,m]) = x[a,m] when = () m < m() by the tree structureof

Sc

B, the cases of the definition of h2 are compatible and the equations are

preserved. So h2 is a homomorphism and even a projection from M onto M2 .

Trivially, we can find a projection h1 from M2 onto M1.

Next note that M1 is a free extension of M0 (a free basis is{y ,c : c C(),m() and <

} {x[a] : a Y1\Y0 and for no < is

a {a() ,m : m [m(), )}}.

So we can find a projection h0 from M1 onto M0. So h0 h1 h0 is a projection

as required. 2.3

2.5 Claim. Theorems 2.2, 2.3 can be strengthened as in 1.3.

2.6 Discussion. Implicit in the proof of 2.3 is an alternative criterion sufficient forthe conclusion of 2.2.


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3 Appendix: characterizing the existencein of an almost free abelian group

To make the main theorem 1.2 more easily read we present part of [Sh 161], moreexactly a variant to [Sh 161],3.6,3.7,p.212.Numbers are as in [Sh 161].

3.1 Definition.

(1) For a regular uncountable cardinal (> 0) we call S a -set if:

(a) S is a set of strictly decreasing sequences of ordinals < .(b) S is closed under initial segments and is nonempty.(c) For S if we let W(, S) =: {i : < i > S} and

(, S) =: Sup W(, S) then whenever W(, S) is not empty, (, S) is a

regular uncountable cardinal and W(, S) is a stationary subset of (, S).Also (,S) = (and by clause (a) we know ( < >, S) ||).

(2) For a -set S, let Sf(= set of final elements of S) be { S : (i) < i >/ S}and Si(= set of initial elements of S) be S\Sf so (Sf = { S : (, S) = 0}).

We sometimes allow = 0. Then the only -set is {}.(3) For -sets S1, S2 we say S1 S2 (S1 a sub--set of S2) if S1 S2 and(, S1) = (, S2) for every S1 (so S1i = S

1 S2i ). Clearly is transitive.

Notation: In this section S will be used to denote -sets.

3.1A Remark. Many of the properties below holds also if we waive the decreasingdemand in clause (a) but not all, and for what we want to analyze we can get suchS, so we have concentrated on this family of sets.

3.2 Claim.

(1) If S is a -set, (, S) > for every Si (holds always for = 0) and G isa function from Sf to then for some S

1 S we have: G is constant on S1f.

(2) If S is a -set and Si, then S[] = { : S} is a (, S)-set, and

(, S[]) = (, S) for every S[].(3) If S is a -set, a regular cardinal, ( S) ((, S) = ) and G is a function

from S to then for some S1 S and < for every S1 we have G() < .(4) If > 0 is regular, W stationary, for W, S

is a -set for some or S

= {} then S =: {} { : S, W} is a -set and

(< > , S) = (, S) for W, S and Si = {}

W

Si .

(5) If S is a -set, F a function with domain S\{}, F() < 1 + then Fis essentially constant for some S1 S which means F { S1 : g() = m} isconstant for each m.(6) For any -set S there is a -set S1 S such that:

(a) all Sf has the same length n

(b) for each < n either

(i) every () ( Sf) is an inaccessible cardinal (not necessarily stronglimit), or

(ii) every() ( Sf) is a singular limit ordinal,


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(c) for each < n, either

(i) ( ( + 1), S) = () for every Sf or

(ii) ( ( + 1), S) = +1S for every Sf (for a fixed +1S ).

(d) The truth value of cf(()) = ( m, S) is the same for all Sf (forconstant , m < n).

Proof. Straightforward, e.g.(5) In first glance we get only: if Si then F { : W(, S)} is constant(by Fodors lemma and the demand W(, S) is a stationary subset of (, S).However, as every S is (strictly) decreasing sequence of ordinals we can iterate

this (simpler if we first apply part (6) clause (a)). 3.2

3.3 Claim. Suppose P is a family of sets which exemplify the failure of P T(, +)(where > ) i.e. a P = |a| , P has no transversal (= one to one choice function) but everyP P of cardinality < has a transversal. Then there is a-set S and function F with domain Sf such that:

(a) For each Sf, F() is a subfamily of P of power .(b) For Si we have (, S) > .(c) For >( + 1), let F0() =: {F() : , S) + ||and if < (, S) is an inaccessible cardinal (> 0) then W(, S) isnot a stationary subset of .

(f) If < > Sf and cf() > then for some natural number k wehave < > k and ( k, S) = cf() (so if is an inaccessiblecardinal then k = g()).

Proof. See [Sh 161].

Remark. Note clause (f), it is crucial; without it we wont be able to prove thedesired conclusion.


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3.4 Definition.

(1) A -system is B : Sc where:

(a) S is a -set, and we let Sc

=: {i : Si

and i < (, S)}(b) Bi Bj when Si and i < j < (, S)(c) If is a limit ordinal < (, S) then B = {Bi : i < }(d) |Bi| < (, S) for i < (, S).

(2) The -system B : Sc is called disjoint if the sets {B(,S) : Si}(see (3) below) are pairwise disjoint.(3) We let Sm =: S\{}, B(,S) =: B

=: {Bi : i < (, S)} for Si.

3.5 Claim. Suppose is a regular uncountable cardinal, B : Sc a -system,

and for Sf, s S we have

() C is a closed unbounded subset of , C = {(, i) : i < cf()}, (, i)increasing continuously with i

() In addition if = , Si, Si, (, S) = cf[()] > 0 then, is a strictly increasing function from (, S) to (, S)

() in clause () if =: () is an inaccessible cardinal (hence necessarilyg() = + 1) then = W(, S) {(, i) : i belong to the range of ,}

1we are interested mainly in the case = 0


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(f)

() If < m < n, Sf, cf[()] = ( m, S) > then

s B()+1\B() where = ((), ,((m))+2); i.e. isthe (((m))+ 2)-th member of C(). Moreover if s

s

= , =

then ((), (m)) = ((), (m)).() If < m < n = g(), Sf, cf[()] = ( m, S) then

s B(+1)\B where = ((), (m)); i.e. is the

((m) + 1)-th member of C(+1) and < () |s\B()| = .

Moreover if s s = , = then

((), (m)) = ((), (m)).

(g) If < g(), Sf, cf[()] then for no < () is s B.(h) For some well ordering : < , < > S} so D : Sc is adisjoint -system, without loss of generality disjont to S.For Sf and = 0, . . . , n 1, we define t

=: D(+1) {A : A F()}.

For Si and (, S) we let

B =

: is a finite sequence, of length 3 + (n g()),

Rang D {} but Rang()

.


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Let

R = (,m,) : Si, lg() = m, lg()and (, S) = cf[()] >

.

For (,m,) R clearly {t : Sf and (m) < } : < (, S)is an increasing continuous sequence of subsets of B which may have cardinality> (, S), each of cardinality < (, S). But B()((),i) : i < (, S) is anincreasing continuous sequence of sets with union B(+1) (remember ((), i) :i < (, S) is an increasing continuous sequence of ordinals with limit () whichhas cofinality (, S)). Hence

E, =:

i < (, S) : i is a limit ordinal such that

{s : Sf} B()((),i)

= {s : Sf and (m) < i

is a club of(, S), so let , : (, S) (, S) be a strictly increasing continuous

function with range E, .It is clear that B : Sc is a disjoint -system (note |Bi| < (, S) as (, S)

is uncountable). Let t = {a(, ,i) : i < } (possibly with repetitions).

We define s by cases:

() if there is m such that < m < lg(), (,m, m) R and ( m, S) >0 (there is at most one such m, and then0 < m,cf(()) = ( m, S) > 0) we let =: (n(), , ((n)) + 1), , m( [ + 1, n)),

t =: {a(, ,j) : j m : m < and m > 0}

()

= 0, , n [ + 1, n), ifcf(()) we let

s =

y0, . . . , y2m1 :m < ,m > 0, for each k < m,

y2k = min{ C() : a(,, 0), . . . ,

a(, ,k) B}and y2k+1 = a(, ,k)

Note that by clause (f) of 3.3, exactly one of those cases occurs.Now B : Sc, s (for Sf, < g()) are as required in 3.6. The least

trivial is (c). Suppose I Sf, |I| < , so {


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20 SAHARON SHELAH

Now we define a function g : Dom g = I, g() = a(, h(), i) : 0 i g().

Clearly g is one-to-one, g()


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REFERENCES

[Sh 161] Saharon Shelah. Incompactness in regular cardinals. Notre Dame Jour-nal of Formal Logic, 26:195228, 1985.

[MkSh 251] Alan H. Mekler and Saharon Shelah. When -free implies strongly-free. In Abelian group theory (Oberwolfach, 1985), pages 137148.Gordon and Breach, New York, 1987. Proceedings of the third confer-ence on Abelian Groups Theory, Oberwolfach.

[EM] Paul C. Eklof and Alan Mekler. Almost free modules: Set theoreticmethods, volume 46 of NorthHolland Mathematical Library. NorthHolland Publishing Co., Amsterdam, 1990.

[EM2] Paul C. Eklof and Alan Mekler. Categoricity results for L,-freealgebras. Annals of Pure and Applied Logic, 37:8199.

[MkSh 366] Alan H. Mekler and Saharon Shelah. Almost free algebras . IsraelJournal of Mathematics, 89:237259, 1995. math.LO/9408213.

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