Remarks on

145 (1994)

Remarks on P _κ λ -combinatorics

by

Shizuo K a m o (Osaka)

Abstract. We prove that { x ∈ P

λ | x ∩ κ is almost x-ineffable} has p

(NIn

)- measure 1 and { x ∈ P

λ | x ∩ κ is x-ineffable} has I-measure 1, where I is the complete ineffable ideal on P

λ. As corollaries, we show that λ-ineffability does not imply complete λ-ineffability and that almost λ-ineffability does not imply λ-ineffability.

In [6], Jech introduced the notion of λ-ineffability and almost λ-inef- fability which are the P _κ λ generalizations of ineffability. Next, Johnson [8]

introduced the notion of complete λ-ineffability. These properties can be characterized by certain ideals on P _κ λ (see [3]). By the definitions, it fol- lows directly that λ-supercompact cardinals are completely λ-ineffable and that λ-ineffable cardinals are almost λ-ineffable. Johnson [8] showed that completely λ-ineffable cardinals are λ-ineffable.

Whether the converse implications also hold seems to be interesting.

Concerning this, Abe [1] proved that almost λ-ineffability and λ-ineffability are equivalent if λ > κ is an ineffable cardinal. It is not difficult to check that complete λ-ineffability does not imply λ-supercompactness. In this paper, we shall prove the following two theorems.

Theorem 4.1. If κ is λ ^<κ -ineffable, then {x ∈ P κ λ | x ∩ κ is almost x-ineffable} ∈ p _∗ (NIn _κ,λ

) ^∗ , where p denotes the projection from P _κ λ ^<κ to P κ λ.

Theorem 4.2. Let I be a normal, (λ ^<κ , 2)-distributive ideal on P κ λ.

Then {x ∈ P _κ λ | x ∩ κ is x-ineffable} ∈ I ^∗ .

By using these theorems, we shall show that λ-ineffability does not im- ply complete λ-ineffability and that almost λ-ineffability does not imply λ-ineffability.

In the proofs of Theorems 4.1 and 4.2, we shall use the notion of strong normality (which was introduced by Carr [4]) and a certain correspondence

1991 Mathematics Subject Classification: Primary 03E55.

[141]

between P κ λ and P κ λ ^<κ . The strong normality and this correspondence will be dealt with in Sections 2 and 3, respectively. The two theorems will be proved in Section 4.

The author got the idea of the correspondence between P _κ λ and P _κ λ ^<κ from discussions with Prof. Y. Abe at Kanagawa University and would like to thank him.

1. Notation and terminology. Throughout this paper, κ denotes a regular uncountable cardinal, and λ a cardinal ≥ κ. Let J be an ideal on a set S. Then J ^∗ denotes the dual filter of J and J ⁺ the set P(S) \ J . For any X ⊂ S, J ⁺ ¹X denotes J ⁺ ∩ P(X). For any f : S → T , f _∗ (J ) denotes the ideal {Y ⊂ T | f ⁻¹ Y ∈ J } on T .

Let A be a set such that κ ≤ |A|. Then P _κ A is the set {x ⊂ A | |x| < κ}.

For each x ∈ P κ A, b x denotes the set {y ∈ P κ A | x ⊂ y & x 6= y}. I κ,A denotes the ideal {X ⊂ P _κ A | X ∩ b y = ∅ for some y ∈ P _κ A}. An element of I ⁺ _κ,A is called unbounded. A subset of P κ A is called club if it is unbounded and closed under unions of increasing chains with length < κ. A subset X of P _κ A is called stationary if X ∩C 6= ∅ for any club subset C of P _κ A. NS _κ,A denotes the ideal {X ⊂ P κ A | X is non-stationary}. A function f from X (⊂ P κ A) to A is called regressive if f (x) ∈ x for all x ∈ X \ {∅}. For any indexed family {X _a | a ∈ A} of subsets of P _κ A, the diagonal union 5 _a∈A X _a and the diagonal intersection 4 _a∈A X a are the sets {x ∈ P κ A | x ∈ X a for some a ∈ x} and {x ∈ P _κ A | x ∈ X _a for all a ∈ x}, respectively. A κ-complete ideal on P _κ A is said to be normal if it contains I _κ,A and is closed under diagonal unions.

A subset X ⊂ P _κ A is said to be A-ineffable, almost A-ineffable, and A-Shelah, respectively, if

∀f _x : x →2 (for x ∈ X) ∃f : A →2 ({x ∈ X | f _x ⊂ f } ∈ NS ⁺ _κ,A ),

∀f x : x →2 (for x ∈ X) ∃f : A →2 ({x ∈ X | f x ⊂ f } ∈ I ⁺ _κ,A ),

∀f _x : x →x (for x ∈ X) ∃f : A →A ∀x ∈ P _κ A ∃y ∈ X ∩ b x (f _y ¹x = f ¹x).

Following Carr [2], [3], define

NIn κ,A = {X ⊂ P κ A | X is not A-ineffable},

NAIn _κ,A = {X ⊂ P _κ A | X is not almost A-ineffable}, NSh κ,A = {X ⊂ P κ A | X is not A-Shelah}.

Carr [2], [3] showed that these are normal ideals on P κ A and that NSh κ,A ⊂ NAIn _κ,A . A cardinal κ is said to be A-ineffable (almost A-ineffable, A- Shelah) if NIn _κ,A (NAIn _κ,A , NSh _κ,A ) is proper.

Let I be an ideal on P κ A and % a cardinal. Then I is said to be (%, 2)-

distributive if for any X ∈ I ⁺ and any family {{X _α,0 , X _α,1 } | α < %} of

disjoint partitions of X, there exist X ⁰ ∈ I ⁺ ¹X and f : % → 2 such that X ⁰ \ X _{α,f (α)} ∈ I for all α < %. Note that this definition is equivalent to the usual definition of (%, 2)- (or (%, %)-)distributivity given in [8]. Following Johnson [8], we say that κ is completely A-ineffable if there exists a proper, normal, (|A|, 2)-distributive ideal on P _κ A. By using the following theorem [8, Theorem 5.1], she proved that completely A-ineffable cardinals are A- ineffable.

Theorem 1.1. For any ideal I on P κ A containing I κ,A , the following statements are equivalent.

(a) I is normal and (|A|, 2)-distributive.

(b) ∀X ∈ I ⁺ ∀f _x : x → 2 (for x ∈ X) ∃f : A → A ({x ∈ X | f _x ⊂ f }

∈ I ⁺ ).

2. Strong normality. From now on, I denotes a proper, κ-complete ideal on P _κ λ containing I _κ,λ . In this section, we shall consider the strong normality of ideals on P κ λ which was introduced by Carr [4]. For x, y ∈ P κ λ, x ≺ y means that x ⊂ y and |x| < |κ ∩ y|. Following Carr [4], I is called strongly normal if

∀X ∈ I ⁺ ∀a _x ≺ x (for x ∈ X) ∃a ∈ P _κ λ ({x ∈ X | a _x = a} ∈ I ⁺ ).

It is clear that strongly normal ideals are normal. Carr [4, Theorems 3.4, 3.5]

showed that, under the assumption that λ ^<κ = λ, the ideals NIn _κ,λ , NAIn _κ,λ and NSh _κ,λ are strongly normal.

For x ∈ P _κ λ, Q _x denotes the set P _κ∩x x (= {t ⊂ x | t ≺ x}). For any indexed family {X _t | t ∈ P _κ λ} of subsets of P _κ λ, 4 _t∈P

_λ X _t denotes the set {x ∈ P κ λ | x ∈ X t for all t ≺ x}, and 5 t∈P

λ X t the set {x ∈ P κ λ | x ∈ X t

for some t ≺ x}. We call 4 _t∈P

_λ X _t and 5 _t∈P

_λ X _t the strong diagonal intersection and union, respectively, of {X _t | t ∈ P _κ λ}. The following lemma is known [5] and can be easily verified.

Lemma 2.1 The following statements are equivalent.

(a) I is strongly normal.

(b) I is closed under strong diagonal unions.

Lemma 2.2. If I is normal and (λ, 2)-distributive, then I is strongly normal.

P r o o f. Let X ∈ I ⁺ and a _x ≺ x for x ∈ X. For each x ∈ X, take β x ∈ x ∩ κ such that |a x | ≤ |x ∩ β x |. Since I is normal, we may assume that β _x = β for all x ∈ X. For each α < λ, set

Y α,0 = {x ∈ X | α ∈ a x }, Y α,1 = {x ∈ X | α 6∈ a x },

W _α = {Y _α,0 , Y _α,1 } ∩ I ⁺ .

Since W α is an I-partition of X for every α < λ, there exist g : λ → 2 and Z ∈ I ⁺ such that

Z ⊂ X and Z \ Y _α,g(α) ∈ I for all α < λ.

Set Y = 4 _α<λ Y _α,g(α) . Since I is normal, Z \ Y ∈ I. So, Y ∈ I ⁺ . Set A = g ⁻¹ {0}. Then it is easy to see that a y = A ∩ y for all y ∈ Y and

|A| ≤ |β|. So, A ∈ P _κ λ. Set Y ₁ = Y ∩ b A. Then Y ₁ ∈ I ⁺ and a _y = A for all y ∈ Y 1 .

Define

S(I) = { 5

t∈P

λ X t ∪ Y | ∀t ∈ P κ λ (X t ∈ I) & Y ∈ I}.

Lemma 2.3. Suppose that κ is an inaccessible cardinal. Then S(I) is the smallest strongly normal ideal containing I.

P r o o f. Since it is clear that S(I) is an ideal, we only verify that S(I) is strongly normal. So, let Y t ∈ S(I) (for t ∈ P κ λ). For each t ∈ P κ λ, take X t,s ∈ I (for s ∈ P κ λ) and A t ∈ I such that Y t ⊂ 5 s∈P

λ X t,s ∪ A t . For each a ∈ P _κ λ, let B _a = S

s,t⊂a X _t,s ∪ A _a . Since κ is inaccessible, B _a ∈ I for all a ∈ P κ λ. It is easy to check that

5

t∈P

λ Y t ⊂ 5

a∈P

λ B a ∪ (P κ λ \ b ω) ∈ S(I).

Corollary 2.4. Suppose that κ is an inaccessible cardinal. Then S(NS κ,λ ) = S(I κ,λ ).

For each τ : P _κ λ → P _κ λ, cl(τ ) denotes the set {x ∈ P _κ λ | x 6= ∅ &

∀t ≺ x (τ (t) ⊂ x)}.

Lemma 2.5. Suppose that κ is an inaccessible cardinal. Let X ⊂ P _κ λ.

Then the following statements are equivalent.

(a) X ∈ S(NS _κ,λ ).

(b) There exists τ : P _κ λ → P _κ λ such that cl(τ ) ∩ X = ∅.

P r o o f. (a)⇒(b). Let X ∈ S(NS _κ,λ ). By the previous corollary, we can take x _a ∈ P _κ λ (for a ∈ P _κ λ) and b ∈ P _κ λ such that

X ⊂ 5

a∈P

λ (P _κ λ \ b x _a ) ∪ (P _κ λ \ bb).

Let τ = hx _a ∪ b ∪ ω | a ∈ P _κ λi. Then cl(τ ) ∩ X = ∅.

(b)⇒(a). Suppose τ : P _κ λ → P _κ λ satisfies cl(τ ) ∩ X = ∅. For each a ∈ P _κ λ, set Y _a = P _κ λ \ τ (a) ^∧ . Let Y = 5 _a∈P

_λ Y _a . Then X ⊂ Y and Y ∈ S(I κ,λ ).

The following lemma is not needed later. However, it seems to be interest-

ing, because if κ is an inaccessible cardinal, then the set X = {x ∈ P _κ λ | x∩κ

is an ordinal and cof(x∩κ) = ω} satisfies {x ∈ X | X∩Q _x ∈ I _κ∩x,x } ∈ NS ⁺ _κ,λ .

Lemma 2.6. Suppose that κ is an inaccessible cardinal. Then {x ∈ X | X ∩ Q _x ∈ I _κ∩x,x } ∈ S(NS _κ,λ ) for any X ⊂ P _κ λ.

P r o o f. To get a contradiction, assume that there exists X ⊂ P _κ λ such that

Y = {x ∈ X | X ∩ Q _x ∈ I _κ∩x,x } ∈ S(NS _κ,λ ) ⁺ .

For each x ∈ Y , take a x ∈ Q x such that b a x ∩ X ∩ Q x = ∅. Since Y ∈ S(NS _κ,λ ) ⁺ , there exists a ∈ P _κ λ such that

Z = {x ∈ Y | a x = a} ∈ S(NS κ,λ ) ⁺ .

Take x, y ∈ Z such that x ≺ y. Then x ∈ X ∩ b a _y ∩ Q _y . A contradiction.

3. A correspondence between P _κ λ and P _κ λ ^<κ . From now on, we assume that κ is an inaccessible cardinal. Let θ = λ ^<κ and p : P _κ θ → P _κ λ denote the projection (i.e., p(y) = y ∩ λ).

Take a bijection h : θ → P _κ λ. Define π = π(h) : P _κ λ → P _κ θ and q = q(h) : P κ θ → P κ λ by

π(x) = h ⁻¹ Q _x for each x ∈ P _κ λ, q(y) = [

h ⁰⁰ y for each y ∈ P κ θ.

Set

C h = {y ∈ P κ θ | ∀α ∈ y (h(α) ≺ q(y)) & q(y) = p(y)}, The following lemma can be easily verified.

Lemma 3.1. (1) qπ(x) = x for any x ∈ b 2 (⊂ P κ λ).

(2) C _h is a club of P _κ θ (so, p ⁰⁰ C _h = q ⁰⁰ C _h is a club subset of P _κ λ).

(3) y ⊂ πq(y) for any y ∈ C _h .

(4) Y ∈ I κ,θ iff π ⁻¹ Y ∈ I κ,λ for any Y ⊂ rang(π).

Lemma 3.2. There exist x ∈ P κ λ and y ∈ P κ θ such that πq¹(rang(π)∩ b y) is the identity function and b x ⊂ q ⁰⁰ (rang(π) ∩ b y).

P r o o f. Take α < θ such that h(α) = 2. Then it is easy to see that πq¹(rang(π) ∩ {α} ^∧ ) is the identity function and b ω ⊂ q ⁰⁰ (rang(π) ∩ {α} ^∧ ).

Corollary 3.3. Let J be an ideal on P _κ θ. If rang(π) ∈ J ^∗ and I _κ,θ ⊂ J , then π ∗ q ∗ (J ) = J .

Lemma 3.4. The following statements are equivalent.

(a) I is strongly normal.

(b) π ∗ (I) is normal.

P r o o f. (a)⇒(b). Assume that I is strongly normal. Let Y α ∈ π ∗ (I) for

α < θ. Set Y = 5 _α<θ Y _α . For each a ∈ P _κ λ, set X _a = π ⁻¹ Y _h

(a) ∈ I. Set

X = 5 a∈P

λ X a . Then π ⁻¹ Y ⊂ X. Since X a ∈ I for all a ∈ P κ λ, it follows that X ∈ I. So, Y ∈ π _∗ (I).

(b)⇒(a). Assume that π _∗ (I) is normal. Let X _a ∈ I for a ∈ P _κ λ. Set X = 5 a∈P

λ X a . For each α < θ, set Y α = π ⁰⁰ X _h(α) . Set Y = 5 α<θ Y α . Since π ⁻¹ Y _α = X _h(α) ∈ I for all α < θ, it follows that Y ∈ π _∗ (I). Since X ⊂ π ⁻¹ Y , we conclude that X ∈ I.

Corollary 3.5. If I is strongly normal, then NS κ,θ ⊂ π ∗ (I). In par- ticular , NS _κ,θ ⊂ π _∗ (S(NS _κ,λ )).

Lemma 3.6. Y ∈ NS _κ,θ iff π ⁻¹ Y ∈ S(NS _κ,λ ) for any Y ⊂ rang(π).

P r o o f. The implication ⇒ follows immediately from the above corollary.

To show the converse, let Y ⊂ rang(π) and X = π ⁻¹ Y ∈ S(NS κ,λ ). By Lemma 2.5, there exists τ : P _κ λ → P _κ λ such that cl(τ ) ∩ X = ∅. Define C ⊂ P _κ θ by

C = {y ∈ C _h | τ (h(α) ∩ λ) ⊂ p(y) for all α ∈ y}.

Then C is a club subset of P _κ θ and C ∩ Y = ∅. So, Y ∈ NS _κ,θ . Lemma 3.7. rang(π) ∈ NSh ^∗ _κ,θ .

P r o o f. To get a contradiction, assume that Y ₀ = P _κ θ\rang(π) ∈ NSh ⁺ _κ,θ . Since C _h is a club, Y = Y ₀ ∩ C _h ∈ NSh ⁺ _κ,θ . Since, for all y ∈ Y , we have y ⊂ π(y ∩ λ) and y 6= π(y ∩ λ), we can take a y (for y ∈ Y ) such that

a y ≺ y ∩ λ and h ⁻¹ (a y ) 6∈ y for any y ∈ Y.

Since κ is θ-Shelah and cof(θ) ≥ κ, by the result of Johnson [8, Cor. 2.7], θ ^<κ = θ. So, NSh _κ,θ is strongly normal. Hence, there is a ∈ P _κ λ such that

Y ⁰ = {y ∈ Y | a _y = a} ∈ NSh ⁺ _κ,θ .

Then h ⁻¹ (a) 6∈ y for all y ∈ Y ⁰ . But this contradicts the fact that Y ⁰ is unbounded in P _κ θ.

Theorem 3.8. Let Y ⊂ P _κ θ and X = q ⁻¹ Y . Then:

(1) Y ∈ NIn ⁺ _κ,θ iff

∀f x : Q x → 2 ( for x ∈ X) ∃f : P κ λ → 2 ({x ∈ X | f x ⊂ f } ∈ S(NS κ,λ ) ⁺ ).

(2) Y ∈ NAIn ⁺ _κ,θ iff

∀f x : Q x → 2 (for x ∈ X) ∃f : P κ λ → 2 ({x ∈ X | f x ⊂ f } ∈ I ⁺ _κ,λ ).

(3) Y ∈ NSh ⁺ _κ,θ iff

∀f _x : Q _x → Q _x (for x ∈ X) ∃f : P _κ λ → P _κ λ such that

∀x ∈ P _κ λ ∃x ⁰ ∈ X ∩ b x (f _x

¹Q _x = f ¹Q _x ).

P r o o f. By Lemmas 3.2 and 3.7, we may assume that Y ⊂ rang(π) and πq¹Y is the identity function.

(1⇒) Let f _x : Q _x → 2 for x ∈ X. Define g _y : y → 2 (for y ∈ Y ) by g y (α) = f _q(y) (h(α)) for any α ∈ y. Since Y ∈ NIn ⁺ _κ,θ , there exists g : θ → 2 such that Y 0 = {y ∈ Y | g y ⊂ g} ∈ NS ⁺ _κ,θ . Set X 0 = q ⁰⁰ Y 0 . By Lemma 3.6, X ₀ ∈ S(NS _κ,λ ) ⁺ . Define f : P _κ λ → 2 by f (t) = g(h ⁻¹ (t)) for all t ∈ P _κ λ.

Then it is easy to see that f _x ⊂ f for all x ∈ X ₀ . So, {x ∈ X | f _x ⊂ f } ∈ S(NS κ,λ ) ⁺ .

(1⇐) Let g _y : y → 2 for y ∈ Y . Define f _x : Q _x → 2 (for x ∈ X) by f _x (a) = g _π(x) (h ⁻¹ (a)) for any a ∈ Q _x . By the hypothesis, there exists f : P κ λ → 2 such that X 0 = {x ∈ X | f x ⊂ f } ∈ S(NS κ,λ ) ⁺ . Set Y 0 = π ⁰⁰ X 0 . By Lemma 3.6, Y ₀ ∈ NS ⁺ _κ,θ . Define g : θ → 2 by g(α) = f (h(α)) for all α < θ.

Then it is easy to see that g _y ⊂ g for all y ∈ Y ₀ . So, {y ∈ Y | g _y ⊂ g} ∈ NS ⁺ _κ,θ . (2), (3) Similar to (1).

Theorem 3.9. The following statements are equivalent.

(a) κ is completely θ-ineffable.

(b) There exists a proper , normal ideal I on P _κ λ which satisfies the (θ, 2)-distributive law.

P r o o f. (a)⇒(b). Assume that (a) holds. Take a proper normal ideal J on P κ θ such that J satisfies the (θ, 2)-distributive law. Set I = q ∗ (J ). Since rang(π) ∈ J ^∗ , I is the desired ideal in (b).

(b)⇒(a). Let I be an ideal on P _κ λ which satisfies (b). Set J = π _∗ (I).

Since I is strongly normal, J is the desired ideal in (a).

4. Theorems. As in the previous section, θ denotes λ ^<κ and p : P _κ θ → P κ λ the projection. In this section, we prove the following theorems.

Theorem 4.1. {x ∈ P _κ λ | x ∩ κ is almost x-ineffable} ∈ p _∗ (NIn _κ,θ ) ^∗ . Theorem 4.2. Let I be a normal, (θ, 2)-distributive ideal on P _κ λ. Then {x ∈ P _κ λ | x ∩ κ is x-ineffable} ∈ I ^∗ .

For Theorem 4.2, in the case of original ineffability, Johnson [7, Cor. 4]

proved a stronger result.

Theorem 4.1 has the following corollary.

Corollary 4.3. Let κ be the least cardinal α such that α is almost α ⁺ -ineffable. Then κ is not κ ⁺ -ineffable.

P r o o f. To get a contradiction, assume that κ is κ ⁺ -ineffable. By a result of Johnson [8], (κ ⁺ ) ^<κ = κ ⁺ . So, p ∗ (NIn _κ,κ

) is proper. Since {x ∈ P κ κ ⁺ |

|x| = (x ∩ κ) ⁺ } ∈ p _∗ (NIn _κ,κ

) ^∗ , by Theorem 4.1, there exists x ∈ P _κ κ ⁺ such

that x ∩ κ is almost x-ineffable and |x| = (x ∩ κ) ⁺ . Since x ∩ κ < κ, this

contradicts the choice of κ.

By using a similar argument, the next corollary follows from Theo- rem 4.2.

Corollary 4.4. Let κ be the least cardinal α such that α is α ⁺ -ineffable.

Then κ is not completely κ ⁺ -ineffable.

First we prove Theorem 4.1. Before starting the proof, we show the following lemma.

Let h : θ → P _κ λ be a bijection, π = π(h), and q = q(h).

Lemma 4.5. Let X ∈ q ∗ (NIn κ,θ ) ⁺ and, for each t ∈ P κ λ, W t be a family of disjoint subsets of X such that |W _t | < κ and X \ S

W _t ∈ I _κ,λ . Then there exists σ ∈ Q

t∈P

λ W _t such that 4

t∈P

λ

σ(t) ∈ S(NS κ,λ ) ⁺ . P r o o f. Take an enumeration hA _s | s ∈ P _κ λi of S

t∈P

λ W _t . For each x ∈ X, define f x : Q x → 2 by

f _x (s) =

 0 if x ∈ A _s , 1 if x 6∈ A s . By Theorem 3.8(3), there exists f : P _κ λ → 2 such that

Z = {x ∈ X | f _x ⊂ f } ∈ S(NS _κ,λ ) ⁺ .

Claim 1. ∀t ∈ P κ λ ∀A ∈ W t (Z \ A ∈ NS ⁺ _κ,λ ⇒ Z ∩ A ∈ NS κ,λ ).

P r o o f. Let t ∈ P κ λ and A ∈ W t and Z \ A ∈ NS ⁺ _κ,λ . Take s ∈ P κ λ such that A = A _s . Take x ∈ P _κ λ such that s ∈ Q _x . Then, since Z \ A ∈ NS ⁺ _κ,λ , we have (Z \ A) ∩ b x 6= ∅. So, f (s) = 0. Hence, Z ∩ A ∩ b x = ∅.

Claim 2. ∀t ∈ P κ λ ∃! A ∈ W t (Z \ A ∈ NS κ,λ ).

P r o o f. Let t ∈ P _κ λ. The uniqueness follow from the assumption that W _t is disjoint. The existence follows from Claim 1 and the fact that Z ∩ S

W _t ∈ S(NS κ,λ ) ⁺ .

By Claim 2, take σ ∈ Q

t∈P

λ W _t such that Z \ σ(t) ∈ NS _κ,λ for any t ∈ P _κ λ. Then σ is as required.

P r o o f o f T h e o r e m 4.1. To get a contradiction, assume that X = {x ∈ P _κ λ | x ∩ κ is not almost x-ineffable} ∈ p _∗ (NIn _κ,θ ) ⁺ . Without loss of generality, we may assume that qπ¹X is the identity function on X and p¹π ⁰⁰ X = q¹π ⁰⁰ X. For each x ∈ X, take f _t ^x : t → 2 (for t ∈ Q x ) such that

∀f : x → 2 ({t ∈ Q _x | f _t ^x ⊂ f } ∈ I _κ∩x,x ).

For each t ∈ P κ λ, define A t (e) (for e ∈ ^t 2) by A t (e) = {x ∈ X | t ∈ Q x

& f _t ^x = e}, and set W _t = {A _t (e) | e ∈ ^t 2}. By Lemma 4.5, there exists σ ∈ Q

t∈P

λ W _t such that Z = 4

t∈P

λ

σ(t) ∈ S(NS κ,λ ) ⁺ .

For each t ∈ P _κ λ, take e _t ∈ ^t 2 such that σ(t) = A _t (e _t ). Then

∀x ∈ Z ∀t ∈ Q x (f _t ^x = e t ).

Since X ∈ p _∗ (NIn _κ,θ ) ⁺ ⊂ NIn ⁺ _κ,λ , there exists e : λ → 2 such that X ⁰ = {x ∈ X | e x ⊂ e} ∈ NS ⁺ _κ,λ . Take τ : P κ λ → P κ λ such that

∀t ∈ P _κ λ ∃s ∈ X ⁰ (t ⊂ s ≺ τ (t) ∈ X ⁰ ).

Since Z ∈ S(NS κ,λ ) ⁺ , there is x ∈ Z such that x ∈ cl(τ ). Set f = e¹x. Then it is easy to see that {t ∈ Q _x | f _t ^x ⊂ f } ∈ I ⁺ _κ∩x,x . But this contradicts the choice of {f _t ^x | t ∈ Q _x }.

Next, we shall prove Theorem 4.2. The following lemma is an analogue of a result of Johnson [8, Theorem 5.1] and can be proved by a similar argument. But for the convenience of the reader, we give a proof.

Lemma 4.6. The following statements are equivalent.

(a) I is normal and satisfies the (θ, 2)-distributive law.

(b) Whenever X ∈ I ⁺ and A _x ⊂ Q _x (for x ∈ X), there exists A ⊂ P _κ λ such that {x ∈ X | A ∩ Q x = A x } ∈ I ⁺ .

P r o o f. (a)⇒(b). For each t ∈ P κ λ, set

X _t,0 = {x ∈ X | t ∈ A _x }, X _t,1 = {x ∈ X | t 6∈ A _x }, W _t = {X _t,0 , X _t,1 }.

Take g : P κ λ → 2 and Z ∈ I ⁺ such that Z \X _t,g(t) ∈ I for each t ∈ P κ λ. Set A = g ⁻¹ {0} and Z ₁ = 4 _t∈P

_λ X _t,g(t) . It is easy to check that A ∩ Q _x = A _x for all x ∈ Z 1 . Since I is strongly normal, Z \ Z 1 ∈ I. So, Z 1 ∈ I ⁺ .

(b)⇒(a). Normality can be easily proved. So, we must only show dis- tributivity. Suppose that X ∈ I ⁺ and W t is an I-partition of X with

|W _t | ≤ 2, for each t ∈ P _κ λ. Without loss of generality, we may assume that W _t = {X _t,0 , X _t,1 } is a disjoint partition of X for all t ∈ P _κ λ. For each x ∈ X, define A x = {t ∈ Q x | x ∈ X t,0 }. By (b), there exists A ⊂ P κ λ such that

X ⁰ = {x ∈ X | A ∩ Q _x = A _x } ∈ I ⁺ . Define g : P _κ λ → 2 by

g(t) =

 0 if x ∈ A,

1 if x 6∈ A.

We claim that X ⁰ \ X _t,g(t) ∈ I for all t ∈ P κ λ. So, let t ∈ P κ λ. Take x ∈ P κ λ such that t ∈ Q _x . Then it is easy to check that (X ⁰ \ X _t,g(t) ) ∩ b x = ∅. Hence, X ⁰ \ X _t,g(t) ∈ I _κ,λ ⊂ I.

Lemma 4.7. Suppose that I is (θ, 2)-distributive. Then {x ∈ X | X ∩ Q _x ∈ NS _κ∩x,x } ∈ I for any X ⊂ P _κ λ.

P r o o f. To get a contradiction, suppose that there exists X ⊂ P κ λ such that

X 0 = {x ∈ X | X ∩ Q x ∈ NS κ∩x,x } ∈ I ⁺ .

For each x ∈ X 0 , take C x ⊂ Q x such that C x is club in Q x and C x ∩ X

∩ Q _x = ∅. Since I satisfies the (θ, 2)-distributive law, by Lemma 4.6 there is D ⊂ P κ λ such that

X 1 = {x ∈ X 0 | C x = D ∩ Q x } ∈ I ⁺ .

Then D is club in P κ λ. So, take t, x ∈ D ∩ X 1 such that t ≺ x. Then t ∈ D ∩ Q _x = C _x . But this contradicts the fact that C _x ∩ X ∩ Q _x = ∅.

P r o o f o f T h e o r e m 4.2. To get a contradiction, assume that X = {x ∈ P κ λ | x ∩ κ is not x-ineffable} ∈ I ⁺ .

For each x ∈ X, take f _t ^x : t → 2 (for t ∈ Q _x ) such that

∀f : x → 2 ({t ∈ Q _x | f _t ^x ⊂ f } ∈ NS _κ∩x,x ).

For each t ∈ P _κ λ, define A _t (g) ⊂ P _κ λ (for g ∈ ^t 2) by A _t (g) = {x ∈ X | t ∈ Q _x & f _t ^x = g} and set W _t = {A _t (g) | g ∈ ^t 2}∩I ⁺ . Since W _t is an I-partition of X for all t ∈ P κ λ, there exist σ ∈ Q

t∈P

λ W t and X 0 ∈ I ⁺ such that X 0 ⊂ X and X 0 \ σ(t) ∈ I for all t ∈ P κ λ.

Set X ₁ = 4 _t∈P

_λ σ(t). Since I is strongly normal, X ₁ ∈ I ⁺ . For each t ∈ P _κ λ, take g _t : t → 2 such that σ(t) = A _t (g _t ). Since X ₁ ∈ I ⁺ , there exists g : λ → 2 such that

X 2 = {x ∈ X 1 | g x ⊂ g} ∈ I ⁺ . By Lemma 4.7,

X 3 = {x ∈ X 2 | X 2 ∩ Q x ∈ NS ⁺ _κ∩x,x } ∈ I ⁺ .

Take x ∈ X 3 . Then it is easy to check that X 2 ∩ Q x ⊂ {t ∈ Q x | f _t ^x ⊂ g¹x}.

So, {t ∈ Q _x | f _t ^x ⊂ g¹x} ∈ NS ⁺ _κ∩x,x . But this contradicts the choice of {f _t ^x | t ∈ Q x }.

References

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[4] —, P

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[5] D. C a r r and D. P e l l e t i e r, Towards a structure theory for ideals on P

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282.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSAKA PREFECTURE GAKUEN-CHOU, SAKAI, JAPAN

E-mail: KAMO@CENTER.OSAKAFU-U.AC.JP

Received 10 March 1993;

in revised form 15 February 1994