17

A note on strictly stable generic structures
Koichiro Ikeda *Faculty of Business Administration
Hosei University

Abstract

We show that there is a generic structure in a finite language suchthat the theory is strictly stable and not \omega ‐categorical, and has finite
closures.

1

The class K

It is assumed that the reader is familiar with the basics of genericstructures. For details, see Baldwin‐Shi [1] and Wagner [3].
Let R, S be binary relations with irreflexivity, symmetricity andR\cap S=\emptyset . Let L=\{R, S\}.
Definition 1.1 Let K_{0} be the class of finite L ‐structures A with thefollowing properties:

1. A\models R(a, b) implies that a, b are not S ‐connected;
2. If A\models R(a, b)\wedge R(b, c) , then a, c are not S ‐connected;
3. If A\models R(a, b)\wedge R(b', c) and b, b' are S ‐connected, then a, c are
not S ‐connected;

4.

A has no S ‐cycles.

Definition 1.2 Let*A\in K_{0}.

and b are S ‐connected.

\bulletFor a, b\in A,\bullet

For a\in A , let a_{E}=a/E , and let A_{E}=\{a_{E}:a\in A\}.

aEb means thataThe author is supported by Grants‐in‐Aid for Scientific Research (No.

17K05350 ).

18\bullet

A binary relation R_{E} on A_{E} is defined as follows: for any a, b\in
A, A_{E}\models R_{E}(a_{E}, b_{E}) iff there are some a', b'\in A with a’Ea, b’Eb
and A\models R(a', b') . By Definition 1.1, the structure A_{E}=(A_{E}, R_{E}) can be considered as an R ‐structure (or an R ‐graph)with irreflexivity and symmetricity.
Notation 1.3 Let

A\in K_{0}.

\bulletLet s(A) denote the number of the S ‐edges in A.

\bulletLet\bullet

Let r(A) denote the number of the R ‐edges in A.

\bulletFor \alpha with 0<\alpha\leq 1 , let \delta(A)=x(A)-\alpha\cdot r(A) .

x(A)=|A|-s(A) .

Definition 1.4 Let

A, B, C\in K_{0}.

\delta(B/A) denote \delta(BA)-\delta(A) .

\bulletLet\bullet

For A\subset B,

A is said to be closed in B ,denoted by A\leq B , if\delta(X/A)\geq 0 for any X\subset B-A.
\bullet\bullet

For A=B\cap C, B and C are said to be free over A , denoted by
B\perp {}_{A}C , if R^{B\cup C}=R^{B}\cup R^{C} and S^{B\cup C}=S^{B}\cup S^{C}.
When B\perp {}_{A}C , we write B\oplus_{A}C for an L ‐structure B\cup C.

Lemma 1.5 (K_{0}, \leq) has the free amalgamation property, i.e., when‐A\leq B\in K_{0}, A\leq C\in K_{0} and B\perp {}_{A}C then B\oplus_{A}C\in K_{0}.

everProof.

Let D=B\oplus_{A}C . We have to check that D satisfies con‐ditions 1‐4 in Definition 1.1. Here, for simplicity, we see condition 2
in Definition??. Take any a, b, c\in D with R(a, b)\wedge R(b, c) . If abc
is contained in either B or C , then it is clear that a and c are not
S‐connected. So we can assume that a\in B-A, b\in A and c\in C-A.
Suppose for a contradiction that a and c are S ‐connected. Then there
is some d\in A with R(d, c) . So \delta(c/A)\leq 1-(\alpha+1)<0 , and hence
A\not\leq C , a contradiction. Hence a and c are not S ‐connected.

Remark 1.6 In [2], Hrushovski proved that there were an \alpha\in(0,1)
and a function

f : \mathbb{N}arrow \mathbb{R} such that1.

f(0)=0, f(1)=1 ;2.

f is unbounded and convex;

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3.

f'(n) \leq\min { r :each n\in\omega.

r= \frac{p-q\alpha}{m}>0, m\leq n andm,p, q\in\omega }forDefinition 1.7 For a function f in Remark 1.6, let K=\{A\in K_{0} :\delta(A')\geq f(x(A')) for any A'\subset A }.
Lemma 1.8 (K, \leq) has the free amalgamation property.
Proof.

Let A, B, C\in K be such that A\leq B, A\leq C and B\perp {}_{A}C.

Let D=B\oplus_{A}C . We want to show that D\in K . By Lemma 1.5, wehave D\in K_{0} . So it is enough to see that f(|D|)\leq\delta(D) . Without lossof generality, we can assume that \delta(C/A)\geq\delta(B/A) . By Remark??,\frac{\delta(B)-\delta(A)}{|B|- A|}\geq f'(|B|)\delta(B)\geq f(|B|)

we havehave. On the other hand, since B\in K , we. Hence we have

Definition 1.9\bullet\delta(D)\geq f(|D|) .

Let \overline{K} denote the class of L ‐structure A satis‐fying A_{0}\in K for every finite A_{0}\subset A.
\bulletFor A\subset B\in\overline{K}, A\leq B is defined by A\cap B_{0}\leq B_{0} for any finite
B_{0}\subset B.

\bulletFor A\subset B\in\overline{K} , we write c1_{B}(A)=\cap\{C:A\subset C\leq B\}.

\bullet

It can be checked that there exists a countable L ‐structure

Msatisfying
1. if M\in\overline{K} ;2. if A\leq B\in K and A\leq M , then there exists a copy B' of B
over A with B'\leq M ;3. if A\subset finM , then c1_{M}(A) is finite.
This2

M is calleda(K, \leq) ‐generic structure.

Theorem

In what follows, let M be the (K, \leq) ‐generic structure, T=Th(M)and \mathcal{M} a big model of T.
Lemma 2.1
is finite.

T has finite closures, i.e., for any finite A\subset \mathcal{M},c1_{\mathcal{M}}(A)20
Proof. For each t\in R , let H_{t}=\{(x, y) : x, r\in\omega, y=x-\alpha r, f(x)\leqy\leq t\} . Since f is unbounded, each H_{t} is finite. Hence any A\subset fin\mathcal{M}has finite closures.

Lemma 2.2

Proof.

Let a_{0}, a_{1} ,Since a_{0}a_{1}thatT is not \omega ‐categorical.

be vertices with the relations S(a_{0}, a_{1}), S(a_{1}, a_{2}),\ldots.

\in\overline{K} , we can assume that a_{0}a_{1}tp(a_{0}a_{n})\neq tp(a_{0}a_{m})\subset \mathcal{M} . It can be checked
S_{2}(T) isfor each distinct m, n\in\omega . Theninfinite. Hence T is not \omega ‐categorical.
For A\subset fin\mathcal{M} and n\in\omega, A is said to be n ‐closed, if \delta(X/A)\geq 0
for any X\subset \mathcal{M}-A with |X|\leq n.
A\leq fin\mathcal{M} and n\in\omega.

Notation 2.3 Let\bulletcltp_{n}(A)=\{X\cong A\}\cup { X is n ‐closed}\bulletcltp (A)= \bigcup_{i\in\omega}cltp_{i}(A)\bullet

E(A)=\{B\in K:A\leq B\}\bulletE^{+}(A)= { B\in E(A) : there is a copy of B over A in \mathcal{M} }\bulletE^{-}(A)=E(A)-E^{+}(A)\bulletptp(A)=\{\exists Y(XY\cong AB) : B\in E^{+}(A)\}\bulletntp(A)=\{\neg\exists Y(XY\cong AB) : B\in E^{-}(A)\}\bulletgtp(A)=cltp(A)\cup ptp(A)\cup ntp(A)\bulletgtp_{n}(A)=cltp_{n}(A)\cup ptp(A)\cup ntp(A)A\subset B\in K_{0} . Then B_{A} is an L\cup\{R_{E}, S_{E}\} ‐structure with the following properties:
Definition 2.4 Let

1. the universe is \{b_{E}:b\in B-A\}\cup A ;2. the restriction of B on A is the L ‐structure A ;3. for a\in A andb\in B-A, B_{A}\models R_{E}(a, b_{E}) iff there is a b'\in B-Awith b’Eb and B\models R(a, b') , and B_{A}\models R_{E}(b_{E}, a) iff there is ab'\in B-A with b’Eb and B\models R(b', a) ;4. for a\in A andb\in B-A, B_{A}\models S_{E}(a, b_{E}) iff there is a b'\in B-A

with b’Eb and B\models S(a, b') , and B_{A}\models S_{E}(b_{E}, a) iff there is ab'\in B-A with b’Eb and B\models S(b', a) ;21
21b, c\in B-A, B_{A}\models R(b_{E}, c_{E}) iff there are b', c'\in B-A with

5. forb’Eb, c’Ec and B\models R(b', c') .
Note 2.5 By the similar argument as in Definition 1.2, the structure
B_{A} is canonically considered as an L ‐structure.
Lemma 2.6 Let A\leq fin\mathcal{M} andgenerated.

n\in\omega .

Then gtp_{n}(A) is finitely

Proof.

Take a sequence (S_{i})_{i\in\omega} of finite subsets of gtp_{n}(A) with
S_{0}\subset S_{1}\subset. . . and \cup S_{i}=gtp_{n}(A) . For i\in\omega , let \sigma_{i}(X)=\wedge S_{i}.

We can assume that \models\sigma_{i}(A') implies A'\cong A . Since f is unbounded,

C_{i}=\{C_{A}', : M\models\sigma_{i}(A'), C'=c1_{M}(A')\} is finite.
i_{0}\in\omega such that

gtp_{n}(A)So there is some

C_{j}=C_{i_{0}} for every j>i_{0} . Hence S_{i_{0} generates.

Lemma 2.7 If gtp(A)=gtp(B) and A\leq C\leq fin\mathcal{M} , then there is a
D withProof.

gtp(AC)=gtp(BD) .
Let\Sigma(XY)=gtp(AC) and let \Sigma_{n}(XY)=gtp_{n}(AC) forn\in\omega . We want to show that \Sigma(BY) is consistent. To show this, itis enough to see that \Sigma_{n} (BY) is consistent for each n . On the other
hand, by Lemma 2.6, \Sigma_{n}(XY) can be considered as some formula\sigma(XY) . So we want to show that \sigma(BY) has a realization. For this,
we prove that \sigma(XY)\wedge\phi(X) has a realization for each \phi(X)\in tp(B) .
Let \tau(X)=\sigma(XY)|x . Note that \tau(X)\wedge\phi(X)\in tp(B) and \tau(X)\vdash
gtp_{n}(A)=gtp_{n}(B) . Take B'\models\tau\wedge\phi in M . Take A'C'\models\sigma in M
with A'Uc1(A')\cong B'Uc1(B') . Let DE be such that DE\cup c1(B')\congC'cl(C')\cup c1(A') . By genericity, we can assume that E\leq M . Then
we have \models\sigma(B'D) , and hence \sigma(XY)\wedge\phi(X) has a realization.
Corollary 2.8 Let A\leq fin\mathcal{M} . Then gtp(A)\vdash tp(A) .
Definition 2.9 Let

A, B, C\subset \mathcal{M} with A=B\cap C . Then the notation

B\downarrow_{A}^{*}C is defined as follows: for each n\in\omega and A^{*}B^{*}C^{*}\models gtp_{n}(ABC)inM,1.

c1(B^{*})nc1(C^{*})=c1(A^{*}) ;2.

c1(B^{*})\perp_{c1(A^{*})}c1(C^{*}) .

Lemma 2.10 Let

A\leq B\leq \mathcal{M}, A\leq E\leq \mathcal{M} and E\downarrow_{A}^{*}B .

gtp (E/A)\vdash gtp (E/B) .

Then

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Proof. For simplicity, we assume that A, B and E are finite. Take
any E_{1}\models gtp(E/A) with E_{1}\downarrow_{A}^{*}B in M . Fix any n . Then there
are realizations E^{*}A^{*}, E_{1}^{*}A^{*}\models gtp_{n}(EA) in M withc1(E^{*})\cong_{c1(A^{*})}
c1(E_{1}^{*}) . Since E\downarrow_{A}^{*}B and E_{1}\downarrow_{A}^{*}B , there is B^{*}A^{*}\models gtp_{n}(BA) withc1(E^{*})\cong_{c1(B^{*})}c1(E_{1}^{*}) . Hence E_{1}\models gtp(E/B) .
Lemma 2.11

T is strictly stable.

Proof.

Let N\prec \mathcal{M} with |N|=\lambda . Take any e\in \mathcal{M}-N . Then thereA\leq N with e\downarrow_{A}^{*}N . Let E=c1(eA) . We can assume
that E\cap N=A . We want to show that gtp(E/A)\vdash gtp(E/N) . Take
any E_{1}, E_{2}\models gtp (E/A) with E_{i}\downarrow_{A}^{*}N . Take any countable N_{0}\leq N.
Take E_{i}^{*}A^{*}\subset M such that E_{1}^{*}A^{*}, E_{2}^{*}A^{*}\models gtp_{n}(EA) and c1(E_{1}^{*}A^{*})\cong
c1(E_{2}^{*}A^{*}) . Hence gtp(E_{1}/N)=gtp(E_{2}/N) . It follows that |S(N)|\leq
is a countable2^{\omega}\cdot\lambda^{\omega}=\lambda^{\omega} . Hence T is stable.

Theorem 2.12 There is a generic structure
properties:

Mwith the following

1. the language is finite;

2. Th (M) is not \omega ‐categorical;
3. Th (M) has finite closures;
4. Th (M) is strictly stable.

References[1] J. T. Baldwin and N. Shi, Stable generic structures. Ann. Pure
Appl. Log. 79, 1 ‐35 (1996)[2] E. Hrushovski, A stable \aleph_{0} ‐categorical pseudoplane. preprint(1988)[3] F. O. Wagner, Relational structures and dimensions. In: Auto‐morphisms of First‐Order Structures, 153‐181, Clarendon Press,Oxford (1994)
Faculty of Business Administration
Hosei UniversityTokyo 102‐8160, JapanE‐mail: ikeda@hosei.ac.jp