数理解析研究所講究録 1213 巻 2001 年 50-54

50

On quasi-minimal structures
ITAI Masanori, WAKAI Kentaro
昌典, 若井板井健太郎

Department of Mathematical Sciences, Tokai University東海大学

理学部

情報数理学科

Introduction

1, we do not know hardly anything
Unlike the model thory of. This situation is very differabout the model thoery ofent from the one concerning the model theory ofor of, where An $([0, 1])=\{f|f$ :is analytic.
for some open$(\mathrm{C}, +, \cdot,0,1)$$(\mathrm{C}, +, \cdot, \mathrm{e}\mathrm{x}\mathrm{p}, 0,1)$$(\mathrm{R}, +, \cdot, <,\mathrm{e}\mathrm{x}\mathrm{p}, 0,1)$$(\mathrm{R}, +, \cdot, <, 0,1, f)_{f\in An([0,1])}$$U$$Uarrow \mathrm{R}$$\supset[0,1]^{n}\}$First attempts to investigate the model theory of
aremade by B. Zil’ber who has conjectured that the structure is aquasi-minimalstructure which is ageneralization of minimal structures.
$(\mathrm{C}, +, \cdot, \mathrm{e}\mathrm{x}\mathrm{p}, 0,1)$

Definition 1. An uncountable structue is called $quasi- \min$ imal if its definable sets are at most countable or c0-countable.
,The conjecture has not yet been answered neither affirmatively nor negatively. As aminor contribution to this line of research we study basic
properties of quasi-minimal structures. It is well known that we can define a
combinatrial geometry on minimal structures using aclosure operation. It isthen very natual to define asimilar geometry on quasi-minimal structures.
We thank TSUBOI Akito of the University of Tsukuba for his valuablecomments and remarks

512Pre-GeometryIn this note we only work with countable languages
the reader is familiar with basic model theory.

$L$. We also assume that

Let $M$ be an uncountable structure and $A\subset M$ . Then

Definition 2.
$=$$\mathrm{c}\mathrm{c}1_{M}(A)${ $b\in M$ :$b$$\models\varphi$,$\varphi^{M}$

is countable for some $\varphi\in L(A)$ }We omit the subscript $M$ if it is clear from context.

Definition 3. Let $X$ be aset and cl be afunction from $P(X)$ to $P(X)$ ,where $P(X)$ denotes the set of all subsets of $X$ . If $X$ and the function clis apre-geometry. Let
satisfy the following properties, we say that$(X, \mathrm{c}1)$$A\subset X$

(I)

$A$

and , $c\in X$ .
$b$$\subset \mathrm{c}1(A)$.

(II) (Finite Character)$b$$\in \mathrm{c}1(A)\Rightarrow b$(III) (Transfer Property)$\in \mathrm{c}1(A_{0})$$\mathrm{c}1(\mathrm{c}1(A))=\mathrm{c}1(A)$(IV) (Exchange Property)$b$$\in \mathrm{c}1(Ac)$for some finite

Proof:.

.

$-\mathrm{c}1(A)\Rightarrow c\in cl(A6)$

Let $M$ be an uncountable structure. We first show thatthese properties under some conditions.

For any infinite structure $M$ ,Proposition 4.

$A_{0}\subset A$$(M, \mathrm{c}\mathrm{c}\mathrm{l})$.
$(M, \mathrm{c}\mathrm{c}\mathrm{l})$satisfies

satisfies (I) and (II).

Clear by the definition of ccl since the language is countable.

Lemma 5.
$|M|$ and,$b$

Suppose $M$ is aquasi-minimal structure. Let $A\subset M$ , $|A|<$.
. Then$\mathrm{t}\mathrm{p}(b/A)=\mathrm{t}\mathrm{p}(c/A)$$c\in M-\mathrm{c}\mathrm{c}1(\mathrm{A})$, then there is aformula $\varphi(x)\in L(A)$ such
Proof: If
hold. By the quasi-minimality of $M$ , either orandthat bothis countable. Hence either or is in $ccl(A)$ . This contradicts to theassumption on , .
$\mathrm{t}\mathrm{p}(b/A)\neq \mathrm{t}\mathrm{p}(c/A)$$\neg\varphi(c)$$\varphi(b)$$\varphi$$b$$\neg\varphi$$b$$c$$c$

52Proposition 6. Assume that M is quasi-minimal and homogeneous. Then(M,ccl) satisfies the transfer property (III).
Let $A\subset M$ . We show that. Clearly:)
holds by (I). For the other direction, it is enough to show thatfor finite $A\subset M$ , since:$M$ ,by (II). Assume that there is an element.
, there is an element
. By
Since
Lemma 5, we have
. So by the homogeneity assumption$M$
on , there is an -automorphism of $M$ such that $f(b)=c$ . Since is in, is also in. This contradicts to the assumption on .
Proof:$\mathrm{c}\mathrm{c}\mathrm{l}(\mathrm{c}\mathrm{c}\mathrm{l}(A))=\mathrm{c}\mathrm{c}\mathrm{l}(A)$$\mathrm{c}\mathrm{c}\mathrm{l}(\mathrm{c}\mathrm{c}\mathrm{l}(A))$$\mathrm{c}\mathrm{c}1(A)$$B\subset$$\mathrm{c}\mathrm{c}\mathrm{l}(A)=\mathrm{U}\{\mathrm{c}\mathrm{c}\mathrm{l}(B)$$\mathrm{c}\mathrm{c}\mathrm{l}(\mathrm{c}\mathrm{c}\mathrm{l}(A))\subset \mathrm{c}\mathrm{c}\mathrm{l}(A)$$|B|<\mathrm{N}_{0}\}$$b\in \mathrm{c}\mathrm{c}\mathrm{l}(\mathrm{c}\mathrm{c}\mathrm{l}(A))-\mathrm{c}\mathrm{c}\mathrm{l}(A)$$|\mathrm{c}\mathrm{c}1(A)|\leq|A|+|L|$$c\in M-\mathrm{c}\mathrm{c}1(\mathrm{c}\mathrm{c}\mathrm{l}(A))$$\mathrm{t}\mathrm{p}(b/A)=\mathrm{t}\mathrm{p}(c/A)$$A$$\mathrm{c}\mathrm{c}\mathrm{l}(\mathrm{c}\mathrm{c}\mathrm{l}(A))$

$b$

$f$

$c$

$c$

$\mathrm{c}\mathrm{c}\mathrm{l}(\mathrm{c}\mathrm{c}\mathrm{l}(A))$

Proposition 7. Assume that $M$ is quasi-minimal, homogeneous and $|M|\geq$. Thensatisfies the exchange property (IV).
$\aleph_{2}$$(M, \mathrm{c}\mathrm{c}\mathrm{l})$Proof: By the finite character (II), it is enough to show the exchange property (IV) assuming that $A\subset M$ is finite. Suppose that there are elements, $c\in M$ such that. By Lemma 5, weand. Let. We construct asequence.
such that $=b$ , $b_{1}=c$ and $i<j\Rightarrow tp(b\{bj/A)$. Supposethat we have chosen $b_{j}(j<i)$ .
$b$$\mathrm{h}$$b\in \mathrm{c}\mathrm{c}1(Ac)-\mathrm{c}\mathrm{c}1(A)$$p(x, y)=\mathrm{t}\mathrm{p}(\ /A)$$\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{t}\mathrm{p}(b/A)=\mathrm{t}\mathrm{p}(c/A)$$=\mathrm{t}\mathrm{p}(k/A)$$\mathrm{b}$$(b_{i}).\cdot\leq\omega_{1}$Claim.

$c\not\in \mathrm{c}\mathrm{c}1(Ab)$$\bigcap_{j<i}p(b_{j}, M)\neq\emptyset$.

Proof of claim: Since $\bigcap_{j<:}p(b_{j}, M)=M-\bigcup_{j<i}(M-p(b_{j}, M))$ , it is enoughto show that $M-p(bj, M)$ is countable for each . Let $d\in M-p(b_{j}, M)$ . Then
there is aformula $\varphi(bj, y)\in p(b_{j},y)$ such that
holds. Since 92 $(6, c)$,andis not countable in $M$ by the homogeneityof $M$ . Henceis countable by the quasi-minimality of $M$ and $d\in$$ccl(A6j)$ . So $M-p(bj, M)$ is countable. This completes the proof of Claim.
Now we finish the proof of proposition. Let. By thedefinition of ,. But is uncountable. Thisis acontradiction$j$$\neg\varphi(b_{j}, d)$$\mathrm{t}\mathrm{p}(b/A)=\mathrm{t}\mathrm{p}(b_{j}/A)$$\varphi(b_{j}, y)$$\neg\varphi(b_{j}, y)$$b_{:} \in\bigcap_{j<:}p(b_{j}, M)$$b_{i}$$B=\{b_{i} :i<\omega_{1}\}\subset \mathrm{c}\mathrm{c}1(Ab_{\omega_{1}})$

$B$

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3Some Examples
1. Any strongly minimal structure is quasi-minimal.

2. Let $M$ be uncountable, aunary predicate and$(M, P)$ is quasi-minimal.
$P$$|\mathrm{A}I^{P}|$coutable, then

3. Letbe atheory of an equivalence relation
with infinitely many
infinite equivalent classes. Then has quasi-minimal models such as;
$T$

$E$

$T$

(a) $E$ is an equivalence relation with uncountably many countableequivalence classes.
(b) $E$ is an equivalence relation with one uncountable class and count-able countable classes.

4. There are no quasi-minimal random graphs.

Assume that $(M, R)$ is a-minimal random graph. Let$a\in M$ and ,. By Lemma 5, we may assume without
loss of generality that any element in
is connected to .
$M$
Since
is arandom graph, there is an element $d\in M$ such that$R(d, a)\wedge R(d, b)\Lambda\neg R(d, c)$ holds.
Then, because $R(d, a)$holds. Hence, but $R(d, b)\wedge\neg R(d, c)$ holds as well.
This is acontradiction.
Proof:$\mathrm{q}\mathrm{u}\mathrm{a},\mathrm{s}\mathrm{i}$$b$$c\in M-\mathrm{c}\mathrm{c}1(a)$$M-\mathrm{c}\mathrm{c}1(a)$$a$$d\in \mathrm{c}\mathrm{c}1(a)$$\mathrm{t}\mathrm{p}(b/d)=\mathrm{t}\mathrm{p}(c/d)$.5. (a)$(\omega_{1}, <)$is not quasi-minimal, since the successor points are defin-

able.
(b)

$(\omega_{1}\cross \mathrm{Z}, <)$($<\mathrm{i}\mathrm{s}$the lexicographic order) is quasi-minimal but nothomogeneous.
(c)is quasi-minimal and homogeneous. But the exchange
property (IV) does not holds (since the cardinality is less than$(\omega_{1}\cross \mathrm{Q}, <)$$\aleph_{2})$.

4Remarks
Unlike strongly minial sets, the first-0rder property of quasi-minimal sets
are not easily understood. The reason for this is that for two elementaril

54
equivalent structures $M$ and $N$ , the quasi-minimality of $M$ may or may not
imply the quasi-minimality of $N$ . This forbids us to employ the usual modeltheoretic tools such as compactness arguments.
Another difficulty is that the luck of natural interesting examples. ZiPber’sconjecture on the structuteseems plausible but at this m0ment we do not know how to study the structure. As avery small first step
we notice the following:$(\mathrm{C}, +, \cdot, \mathrm{e}\mathrm{x}\mathrm{p}, 0,1)$

Remark 8. It seems very natural to claim that for aquasi-minimal struc, the expanded structure $(M, \cdots, P_{i}(i\in\omega))$ is also quasi-minimal
ture $(M$ ,is aunary predicate whose interpretation is acountable subwhere each$M$
set of. As acorollary to this we have thatis
quasi-minimal where eachis aunary predicate whose interpretation is acountable subset of C.
$\cdots$ $)$$P_{i}$$(\mathrm{C}, +, \cdot, 0, 1, P_{i}(i\in\omega))$$P_{i}$

Remark 9. In Section 2we studied the basic pre geometric properties ofquasi-minimal sets. Although our proof used the additional homogeneity
assumption on the structure, it is not clear whether this assumption is necessary.

Remark 10. In model theory we often work in asaturated model. It seems
that our usual arguments for constructing saturated models are not enoughto define asaturated quasi-minimal structures.

5References
1. A. J. Wilkie, Liouville Functions, Handwritten Notes, 2000
2. B. Zil’ , Fields with pseudo-exponentiation, preprint, 200$\mathrm{b}\mathrm{e}\mathrm{r}$