数理解析研究所講究録 1469 巻 2006 年 92-95

92

On P-closure operator in quasi-minimal structures
前園 久智 (Hisatomo MAESONO)

早稲田大学メディアネットワークセンター

Media Network Center, Waseda University)
Abstract
Itai, Tsuboi and Wakai investigated the geometric properties ofqusai-minimal structures by using the countable closure [1]. I tried todefine another closure operator in such structures.

1 Quasi-minimal structure and the countable closure
We recall some definitions.

Definition 1 An uncountable structure $M$ is called quasi-minimal if every
definable subset of $M$ with parameters is at most countable or co-countable.
. The n-th countable closure
Let $M$ be an uncountable structure andof is inductively defined as follows :and,i countable}:{(the countable closure of ). We omit theWe put
superscript $M$ if it is clear rom the context.
$A\subseteq M$$\mathrm{c}\mathrm{c}1_{n}^{M}(A)$$A$$\mathrm{c}\mathrm{c}1_{0}^{M}(A)=A$$\mathrm{c}\mathrm{c}1_{n+1}^{M}(A)=\cup$$\phi^{M}$$\mathrm{V}(\mathrm{X})\in L(\mathrm{c}\mathrm{c}1_{n}^{M}(A))$$\phi^{M}$$\mathrm{s}$$A$$\mathrm{c}\mathrm{c}1^{M}(A)=\bigcup_{n\in\omega}\mathrm{c}\mathrm{c}1_{n}^{M}$$\mathrm{f}$And we recall the notion of pregeometry.

Definition 2 Let $X$ be an infinite set and cl a function from $P(X)$ to $P(X)$
where $P(X)$ denotes the set of all subsets of $X$ . If the function cl satisfies
is a pregeometry.
the following properties, we say$(X, \mathrm{c}1)$(I)(II) (Finite character),(III)(IV) (Exchange axiom)$A\subseteq B\Rightarrow A\subseteq \mathrm{c}1(A)\subseteq \mathrm{c}1(B)$,$b\in \mathrm{c}1(A)\Rightarrow b\in \mathrm{c}1(A_{0})$for some finite$A_{0}\subseteq A$$\mathrm{c}1(\mathrm{c}1(A))=\mathrm{c}1(A)$$b\in \mathrm{c}1(A\cup\{\mathrm{c}\})-\mathrm{c}1(A)$The countable closure is a closure operator$\Rightarrow c\in \mathrm{c}1(A\cup\{b\})$

.

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Fact 3 [1] Let $M$ be a quasi-minimal structure. Then it is clear thatsatisfies the first three properties (I) through (III).

$(M, \mathrm{c}\mathrm{c}\mathrm{l})$The exchange axiom (IV) does not hold in general in. Itai etal. showed some conditions for $M$ such thatsatisfies the exchange$(M,\mathrm{c}\mathrm{c}\mathrm{l})$$(M, \mathrm{c}\mathrm{c}\mathrm{l})$axiom.

The notion of quasi-minimal structures is a generalization of minimal structures. Thus the countable closure is the canonical closure operator for quasiminimal structures. However, I considered that the countable closure is divided by some F-closures.

2 Internality, foreignness and P-closureFirst we recall some definitions ffom [5].
of partial types is -invariant if it is invariant
Definition 4 A family
under A-automorphisms, (where is a subset of a sufficiently large saturated
model as usual.)
Let be an A-invarint family of partial types.
A partial type over is -internal if for every realization of , there, types from based on , and realizations of , such that
is
$A$

$P$

$A$

$P$

$B\backslash \mathrm{L}_{A}a$

$a$

$B$

$P$

$\overline{p}$$a\in \mathrm{d}\mathrm{c}1(B\overline{c})$

$P$

$A$

$q$

$\overline{p}$.

A patial type is -analysable if for anyisa)
such that$P$$q$$\mathrm{t}\mathrm{p}(a_{i}/A_{l} a_{j} :i<\alpha)$:, there are (-internal for all $i<\alpha$ , and$a\models \mathrm{g}$$\in \mathrm{d}\mathrm{c}1(A, a)$$a\in \mathrm{b}\mathrm{d}\mathrm{d}(\mathrm{A}, a_{i} :$\mathrm{g}$$\overline{c}$j<\mathrm{i})$$P$.

$\mathrm{i}<$$a_{i}$, and,if for all
foreign toA complete type.
realizations of extensions of types in over , we always have
And let be an -invariant family of types.
A partial type is co- foreign to if every type in is foreign to .
of a set is the collection of all element such that
The closure
is -analysable and foreign to .
$q\in S(A)$ is

$P$

$P$

$a^{\llcorner_{AB}}\backslash \overline{c}$

$P$

$P$

$\mathrm{d}\mathrm{P}(\mathrm{A})$$P$$\mathrm{t}\mathrm{p}(a/A)$$B\backslash [perp]_{A}a$$\emptyset$

$q$

$P$

$B$

$P$

$\overline{c}$$a\models q$$A$$\mathrm{c}\mathrm{o}$

$q$

$a$

$P$

Remark 5 The -analysable assumption could be modified or even omitted,
resulting in a larger P-closure.
$P$

Fact 6 -closure is a closure operator,(III) in Definition 2.
$P$

3

$P$

$\mathrm{i}.e$. it satisfies the axioms (I) and-closure for quasi-minimal structures

We recall another notion from [1]84Definition 7 Let $M$ be quasi-minimal. Then a type $p(x)$ defined by$p(x)=\{\psi(x)\in L(M) : |\psi^{M}|\geqq\omega_{1}\}$is a complete type in $S(M)$ . The type $p(x)$ is called the main type of $M$ .

We define -closures in some particular situation to make good in arguments. They may be defined under more general assumption. From now onin this section, $M$ denotes an -saturated quasi-minimal structure. And its
theory Th(M) is -stable. We recall the next theorem fiom [1],$P$$\omega$$\omega$

Theorem 8 Let $M$ be a quasi-minimal structure. And Th(M) is w-stable.
stucThen $M$ can be elementarily embedded to an -saturated
$M’$.
terre
$quas\mathrm{i}rightarrow minimal$$\omega$

We restricts the argument in a quasi-minimal structure $M$ . So in the next
definitions, any tuple is taken ffom $M$ and any set denotes a finite subset
of $M$ . Moreover we assume that $p(x)\in S(M)$ is the main type of $M$ and$p(x)$ is strongly based on 0.

Definition 9 The set of types is defined by $P=\{p\lceil A : A\subseteq M\}$ .
, there are
A complete type $q\in S(A)$ is -internal (over ) if for any.
such that, andover with
with
$P$

$A$

$P$

$B\supset A$$B\backslash [perp]_{A}a$$B$$\overline{c}\models\overline{r}$$a\models \mathrm{g}$$a\in \mathrm{d}\mathrm{c}1(B\overline{c})$$\overline{r}\lceil A\in P$-analysability is defined in the same way as the original definition from theab ove P-internality.
A complete type $q\in S(A)$ is co- foreign to if for any $b\models r\in P$ over ,, and any realization of an extension of over
any $C\supset A$ with
$P$

$B$

$P$

$a’$

$b\backslash \mathrm{L}_{B}C$$C$,$b\backslash \mathrm{L}_{Bc^{a’}}$.

The -closuresuch that$P$$\mathrm{c}1’ p(B)$$\mathrm{t}\mathrm{p}(a/A)$

is$P$of a (any) set-analysable and$q$

is the collection of all element.
-foreign to for some finite$B\subseteq M$$a$$A\subseteq B$$P$$\mathrm{c}\mathrm{o}$Remark 10 If there were no homogeneity of $M$ , does not have the invarince property.
The above -internality is defined for partial types with respect to in general. However, although a partial type is -internal over , it is not always-internal over :) .
$P$

$P$

$P$

$\pi$

$B$

$P$

$P$

$A$

$A$

We can prove the next fact.
Fact 11 Under the same assumptions as the former definition, the P-closure’
is a closure operator.
satisfies the axioms (I) through (III) inDefinition 2.
.
Andfor$(M, \mathrm{c}1_{P}’)$$\mathrm{c}1_{P}$$\mathrm{a}\mathrm{c}1(A)$$\subseteq \mathrm{c}1_{P}’(A)$$\subseteq \mathrm{c}\mathrm{c}1(A)$$A\subseteq M$

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Next we must proceed to the exchange axiom. I will investigate theconditions under which ’ has the exchange axiom. But I doubt if it issignificant.
It is known that the notions of internality and foreignness are available forthe characterization of stable groups. In particular, there are interpretation
theorems of automorphism groups. I will consider whether such situationarises in quasi-minimal structures.
$\mathrm{c}1_{P}$References[1] M.Itai, ATsuboi and K.Wakai, Construction of saturated quasi-minimal
al structure, J. Symbolic Logic, vol. 69 (2004) pp. 9-22
[2] M.Itai and K.Wakai, -saturated quasi-minimal models of Th(Q\mbox{\boldmath $\omega$}, ,, Math. Log. Quart, vol. 51 (2005) pp. 258-262[3] A. Pillay, Geometric stability theory, Oxford Science Publications, 1996
[4] F.O.Wagner, Stable groups, Cambridge University Press, 1997
[5] F.O.Wagner, Simple theories Kluwer Academic Publishers, 200$\omega$$\sigma)$$+$