75

ON THE AUTOMORPHISM GROUP OF A HRUSHOVSKI’SPSEUDOPLANE ASSOCIATED TO 5/8
HIROTAKA KIKYOGRADUATE SCHOOL OF SYSTEM INFORMATICS

KOBE UNIVERSITY

ABSTRACT. The automorphism group of Hrushovski’s pseudoplane as‐sociated to 5/8 is a simple group.

Hrushovski’s construction, automorphism group, simple group
03C10,03C13,03C25,03C30
1. INTRODUCTION

D. Evans, Z. Ghadernezhad, and K. Tent have shown that the automor‐

phisms groups of certain countable structures obtained using the Hrushovski
amalgamation method are simple groups. Among them, there are genericstructure of K_{f} for certain f with coefficient 1/2 for the predimension func‐
tion. They conjectured that the automorphism group of the generic structureof K_{f} is a simple group if the coefficient of the predimension function for

K_{f} is rational.

In this paper, we show that the automorphism group of Hrushovski’s orig‐
inal pseudoplane associated to a predimension function with coefficient 5/8
is a simple group. Actually, we prove a sufficient condition given by Evans,
Ghadernezhad, and Tent. We are going to treat the general rational cases inanother paper [14].
We essentially use notation and terminology from Baldwin‐Shi [3] andWagner [15]. We also use some terminology from graph theory [4].

For a set X, [X]^{n} denotes the set of all subsets of X of size n , and |X| thecardinality of X.
We recall some of the basic notions in graph theory we use in this paper.

These appear in [4]. Let G be a graph. V(G) denotes the set of vertices of
G and E(G) the set of edges of G. E(G) is a subset of [V(G)]^{2}. |G| denotes|V(G)| . The degree of a vertex v is the number of edges at v . A vertex

76
of degree 1 is a leaf.

G is a path x_{0}x_{1}\ldots x_{k} if

V(G)=\{x_{0},x_{1}, ...,x_{k}\} andE(G)=\{x_{0}x_{1},x_{1}x_{2}, ...,x_{k-1}x_{k}\} where the x_{i} are all distinct. x_{0} and x_{k} areends of G . The number of edges of a path is its length. A path of length 0 is asingle vertex. G is a cycle x_{0}x_{1}\ldots x_{k-1}x_{0} if k\geq 3, V(G)=\{x_{0},x_{1}, . . .,x_{k-1}\}and E(G)=\{x_{0}x_{1},x_{1}x_{2}, ...,x_{k-2}x_{k-1},x_{k-1}x_{0} \} where the x_{i} are all distinct.

The number of edges of a cycle is its length. A non‐empty graph G isconnected if any two of its vertices are linked by a path in G. A connectedcomponent of a graph G is a maximal connected subgraph of G. A forest isa graph not containing any cycles. A tree is a connected forest.
To see a graph G as a structure in the model theoretic sense, it is a struc‐ture in language \{E\} where E is a binary relation symbol. V(G) will be theuniverse, and E(G) will be the interpretation of E . The language \{E\} will
be called the graph language.
Suppose A is a graph. If X\subseteq V(A), A|X denotes the substructure B of A
such that V(B)=X . If there is no ambiguity, X denotes A|X . We usuallyfollow this convention. B\subseteq A means that B is a substructure of A . A sub‐structure of a graph is an induced subgraph in graph theory. A|X is the sameas A[X] in Diestel’s book [4].

We say that X is connected in A if X is a connected graph in the graphtheoretical sense [4]. A maximal connected substructure ofA is a connectedcomponent of A.
Let A, B, C be graphs such that A\subseteq C and B\subseteq C. AB denotes C|(V(A)\cup

V(B)), A\cap B denotes C|(V(A)\cap V(B)) , and A-B denotes C|(V(A)-V(B)) .
If A\cap B=\emptyset, E(A,B) denotes the set of edges xy such that x\in A and y\in B.
We put e(A,B)=|E(A,B)|. E(A,B) and e(A,B) depend on the graph inwhich we are working.
Let D be a graph and A, B , and C substructures of D . We write D=B\otimes_{A}C
if D=BC, B\cap C=A , andE(D)=E(B)\cup E(C) . E(D)=E(B)\cup E(C)means that there are no edges between B-A and C-A. D is called a freeamalgam of B and C overA. If A is empty, we write D=B\otimes C , and D isalso called a free amalgam of B and C.
Definition 1.1. Let \alpha be a real number such that 0<\alpha<1.

(1) For a finite graph A , we define a predimension function \delta by \delta(A)=|A|-\alpha|E(A)|.

(2) Let A and B be substructures of a common graph. Put \delta(A/B)=\delta(AB)-\delta(B).

Definition 1.2. Let A and B be graphs with A\subseteq B , and suppose A is finite.

77
A\leq B if whenever A\subseteq X\subseteq B with X finite then \delta(A)\leq\delta(X) .
A<B if whenever A\subset X\neq-\subseteq B with X finite then \delta(A)<\delta(X) .

We say that A is closed in B if A<B.
A<^{-}B if whenever A\subset X\subset B\proptoarrow with X finite then \delta(A)<\delta(X) .
Let K_{\alpha} be the class of all finite graphs A such that \emptyset<A.
Some facts about < appear in [3, 15, 16]. Some proofs are given in [12].
Fact 1.3.

IfA<B\subseteq D and C\subseteq D then A\cap C<B\cap C.

Fact 1.4. Let D=B\otimes_{A}C.

(1) \delta(D/A)=\delta(B/A)+\delta(C/A) .
(2) IfA<C then B<D.
(3) If A<B and A<C then A<D.

Fact 1.5.

(1) LetA, B, C and D be graphs with D=B\otimes C and A\subseteq D.

Then\delta(D/A)=\delta(B/A\cap B)+\delta(C/A\cap C) .

(2) LetLet \{D_{1},D_{2}, ...,D_{k}\}Dbe the set of all connected components of where the D_{i} are all
D be a graph and A a substructure of D.

distinct. Then

\delta(D/A)=\sum_{i=1}^{k}\delta(D_{i}/A\cap D_{i}).

Let B, C be graphs and g:Barrow C a graph embedding. g is a closed
embedding of B into C if g(B)<C . Let A be a graph with A\subseteq B and A\subseteq C.
g is a closed embedding overA if g is a closed embedding and g(x)=x forany x\in A.

In the rest of the paper, K denotes a class of finite graphs closed underisomorphisms.
Definition 1.6. Let K be a subclass of K_{\alpha}. (K, <) has the amalgama‐tion property if for any finite graphs A,B,C\in K , whenever g_{1} : Aarrow B andg_{2} : Aarrow C are closed embeddings then there is a graph D\in K and closed
embeddings h_{1} : Barrow D and g_{2}:Carrow D such that h_{1}\circ g_{1}=h_{2}\circ g_{2}.
K has the hereditary property if for any finite graphs A,B , whenever A\subseteq
B\in K then A\in K.
K is an amalgamation class if \emptyset\in K and K has the hereditary property

and the amalgamation property.
A countable graph M is a generic structure of (K, <) if the following
conditions are satisfied:

78(1) If A\subseteq M and A is finite then there exists a finite graph B\subseteq M suchthat A\subseteq B<M.

(2) If A\subseteq M then A\in K.

(3) For any A, B\in K , if A<M and A<B then there is a closed embed‐
ding of B into M over A.
Let A be a finite structure of M . There is a smallest B satisfying A\subseteq B<
M , written

c1(A) . The set c1(A) is called the closure of A in M.

Fact 1.7 ([3, 15, 16]). Let (K, <) be an amalgamation class. Then there isa generic structure of (K, <) . Let M be a generic structure of (K, <) . Then
any isomorphism between finite closed substructures of M can be extended
to an automorphism of M.
Definition 1.8. Let K be a subclass of K_{\alpha}. (K, <) has the free amalgama‐tion property if whenever D=B\otimes_{A}C with B,C\in K, A<B and A<C thenD\in K.

By Fact 1.4 (2), we have the following.
Fact 1.9. Let K be a subclass of K_{\alpha} . If (K, <) has the free amalgamation
property then it has the amalgamation property.

Definition 1.10. Let \mathbb{R}^{+} be the set of non‐negative real numbers. Sup‐
pose f : \mathbb{R}^{+}arrow \mathbb{R}^{+} is a strictly increasing concave (convex upward) un‐
bounded function. Assume that f(0)=0 , and f(1)\leq 1 . We assume that f
is piecewise smooth. f_{+}'(x) denotes the right‐hand derivative at x . We havef(x+h)\leq f(x)+f_{+}'(x)h for h>0 . Define K_{f} as follows:
K_{f}=\{A\in K_{\alpha}|B\subseteq A\Rightarrow\delta(B)\geq f(|B|)\}.

Note that if K_{f} is an amalgamation class then the generic structure of(K_{f}, <) has a countably categorical theory [16].
A graph X is normal to f if \delta(X)\geq f(|X|) . A graph A belongs to K_{f} if
and only if U is normal to f for any substructure U of A.
2. THEOREMS BY EVANS, GHADERNEZHAD, AND TENTIn this section, we fix a generic structure M of K_{f} . Many of the following
definitions and facts are by Evans, Ghadernezhad, and Tent[5].
Definition 2.1. Let A\subseteq M. Aut(M/A) denotes the set of automorphisms ofM fixing A pointwise. Let b\in M. orb(b/A) denotes the Aut(M/A) ‐orbit ofb . So, orb (b/A)=\{\sigma(b)|\sigma\in Aut(M/A)\}.

79
Definition 2.2. Let A\subseteq M be finite. The dimension d(A) of A is defined byd(A)=\delta(c1(A)) . Let B\subseteq M be also finite. The relative dimension d(A/B)
is defined by d(A/B)=d(AB)-d(B) .

Definition 2.3. Suppose b\in M and A<M with A finite. We say that b isbasic over A if b\not\in A and whenever A\subseteq C<M and d(b/C)<d(b/A) thenb\in C . In this case, orb (b/A) is called a basic orbit over A.

Definition 2.4. We say that M is monodimensional if for every finite A<M
and basic orbit D over A there is a finite B<M with M=c1(BD) and A\subseteq B.

Definition 2.5. Suppose

A<M and b\in M a single element.

b\perp A if

c1(bA)=bA and d(b/A)=d(b) .
Fact 2.6. Suppose A<M and b_{1},b_{2}\in M be single elements. If b_{1}\perp A andb_{2}\perp A then b_{1} and b_{2} are conjugate overA in M.

Proof. Suppose b_{1}\perp A and b_{2}\perp A . We have c1(b_{1}A)=b_{1}A by the defini‐tion. So, \delta(b_{1}/A)=\delta(b_{1})=1 . This means that there are no edges betweenb_{1} and A . By the same argument, there are no edges between b_{2} and A.
Hence, b_{1}A and b_{2}A are isomorphic over A and also b_{1}A<M and b_{2}A<M.
Therefore, the partial isomorphism between b_{1}A and b_{2}A over A can be\squareextended to an automorphism of M by Fact 1.7.
Fact 2.7. If M=c1(AD) for some finite A<M and a basic orbit D over A
then M is monodimensional.

Fact 2.8. If M is monodimensional then the automorphism group of M is a
simple group.
3. HRUSHOVSKI’S BOUNDARY FUNCTIONSDefinition 3.1 ([7]). Let \alpha be a positive real number. We define x_{n}, e_{n}, k_{n},d_{n} for integers n\geq 1 by induction as follows: Put x_{1}=2 and e_{1}=1 . Assume
that x_{n} and e_{n} are defined. Let r_{n} be a smallest rational number r such that

r=k/d>\alpha with d\leq e_{n} where k and d are positive integers. Let k_{n} andd_{n} be coprime positive integers with k_{n}/d_{n}=r_{n} . Finally, let x_{n+1}=x_{n}+k_{n},and e_{n+1}=e_{n}+d_{n}.

Let a_{0}=(0,0) , and a_{n}=(x_{n},x_{n}-e_{n}\alpha) for n\geq 1 . Let f be a function
from \mathbb{R}^{+} to \mathbb{R}^{+} whose graph on interval [x_{n},x_{n+1}] with n\geq 0 is a line seg‐ment connecting a_{n} and a_{n+1} . We call f a Hrushovski ’s boundary function
associated to \alpha.

80
Fact 3.2 ([7]). Let D=B\otimes_{A} C. Suppose B is normal to f, x_{n}\leq|B|<x_{n+1}and \delta(C/A)=x-e\alpha>0 with positive integers x, e. Ifx/e\geq k_{n}/d_{n} then D
is normal to f.
Fact 3.3 ([7]). Let D=B\otimes_{A} C. If \delta(A)<\delta(B), \delta(A)<\delta(C), andA, B, C
are normal to f then D is normal to f.

Fact 3.4 ([7]). Let f be a Hrushovski s boundary function associated toa.
Then f is strictly increasing and concave, and (K_{f}, <) has the freeamalgamation property. Therefore, there is a generic structure of (K_{f}, <) .
Any one point structure is closed in any structure in K_{f} . If \alpha is rational
then f is unbounded.
In the rest of the paper, we fix \alpha=5/8.
Note that 2 8−3\cdot

Proposition 3.5.

\cdot

5=1 . Therefore,

2-3\alpha=1/8.

(1) Let k\geq 0 be an integer. Whenever 0<x<3+

8(k+1) and y/x>5/8 theny/x\geq(2+5k)/(3+8k) .

(2) We refer to Definition 3.1. Suppose e_{n}\geq 3 . Let l be a largest integer
l' with 3+8l'\leq e_{n} . Then k_{n}=2+5l and d_{n}=3+8l.

Proof. (2) follows from (1). So, we show (1).
First, note that for any integers u, v, 8v-5u=1 if and only if u=3+8k
and v=2+5k with an integer k . Also, (2+5k)/(3+8k) is decreasing onk.

By inspection, whenever 0<x<3+8=11 and y/x>5/8 then y/x\geq

2/3.

Suppose u=3+8k and v=2+5k with k\geq 1 . We have\frac{v}{u}-\frac{5}{8}=\frac{1}{8u}.
Assume u<x<u+8 andy/x>5/8 . Note that x<2u because 8<u . We

have

Therefore,\frac{y}{x}-\frac{5}{8}=\frac{8y-5x}{8x}\geq\frac{2}{8x}>\frac{1}{8u}.
\frac{y}{x}>\frac{v}{u}.

We have (1).

By this proposition, we have a following chart:\square81
81

Also, for n\geq 3 , we havef(x_{n+1})=f(x_{n})+ \frac{1}{8}.
The following are easy.
Lemma 3.6.

(1) Let C=A\otimes_{p}B where p is a single vertex and A,B\in

K_{f} . Then C\in K_{f}.

(2) Any finite forest belongs to K_{f}.
(3) Any cycle of length 6 or more belongs to K_{f}.
Lemma 3.7. Let B=A\otimes_{\{x,y\}}P where P=x\cdots y is a path of length 3 ormore. If the distance of x and y is 3 or more in A andA\in K_{f} then B\in K_{f}.
Lemma 3.8. Let

B=A\otimes_{\{x,y,z\}}P where

V(P)=\{x,y,z,x',y',z',w\} and E(P)= \{xx\prime,x'w,yy',y'w,zz',z'w\}.
If the pairwise distances among x, y, z in A are 2 or more andA\in K_{f} thenB\in K_{f}.

4. SPECIAL STRUCTURES

Let B be a graph with V(B)=\{a_{1},a_{2},b_{1},b_{2}\} and E(B)=\{a_{1}b_{1},a_{2}b_{2},b_{1}b_{2}\}and let A=\{a_{1},a_{2}\} . Then A<B anda_{1}\delta(B/A)=1/8.

a_{2}Suppose that B is a closed subset of M . Then b_{1} and b_{2} are basic over A

because 1/8 is the smallest positive possible dimension.
Let W_{1} be the following graph:

82

Let F_{1} be the set of leaves of W_{1} . Then F_{1}<^{-}W_{1} and \delta(W_{1}/F_{1})=0.

Make a free amalgam of 4 copies of B over A and attaching W_{1} to it, we
get:

The point c belongs to the closure of basic points over A.
Unfortunately, This structure does not belong to K_{f} . But it turns out thatany proper substructures belong to K_{f}.
We can make a “wreath” W_{3} with 3 copies of W_{1} :Let F be the leaves of W_{3} . Then F<^{-}W_{3} and \delta(W_{3}/F)=0 . A cyclewith length 6 belongs to K_{f} . Therefore, W_{3} belong to K_{f}.
Lemma 4.1. Let

C_{1}=(B_{1}\otimes_{A}B_{2}\otimes_{A}\cdots\otimes_{A}B_{9})\otimes_{F}W_{3}where each B_{i} is isomorphic to B overA, F=\{b_{11},b_{12}, ..., b_{19}\} , with b_{1i}\in
B_{i} the isomorphic images of b_{1} , and F is also the set of leaves of W_{3} . Then
C_{1}\in K_{f} . The following is a picture of C_{1} :83

9Proof. Let S be the cycle of length 6 at the top including c_{1} in C_{1} . We canrepresent C_{1} as

C_{1}=H_{1}\otimes_{AS}H_{2}\otimes_{AS}H_{3}\otimes_{AS}H_{4}\otimes_{AS}H_{5} \otimes_{AS}H_{6}\otimes_{AS}H_{7}\otimes_{AS}H_{8}\otimes_{AS}H_{9}where each H_{i} is B_{i}p_{i}S, p_{i} is adjacent to b_{1i} and to a single vertex in S . We
have H_{i}\in K_{f} and AS<^{-}H_{i} for each i.

We have to show that if U\subseteq C_{1} then U is normal tof.

Case U\cap A\neq A . In this case, U\cap AS<U\cap H_{i} for each i . Therefore,

U\in K_{f} by the free amalgamation property.
Case A\subseteq U.
Subcase 1:

U=B_{I}\otimes_{F_{I}}U' where I is a subset of \{1,2, \ldots,9\}, B_{I}=\otimes_{A}\{B_{i}\}_{i\in I}, F_{l}=\{b_{1i}\}_{i\in l} , and U' is a substructure of W_{3}.

Let x be the number of points in U'-F_{I} and e the number of edges in U'.
To show that U is normal to f , we can assume that e\leq 3|I| and x<2|I| by
Lemmas 3.6, 3.7, and 3.8.

Suppose |I|\leq 4 . Then |B_{l}|=x_{2+|I|} , and e\leq 3|I|=e_{2+|I|} . Since x-e\alpha=\delta(U'/F_{I})>0 , we have x/e>\alpha . Hence, x/e\geq k_{2+|I|}/d_{2+|I|} . Therefore, U'

is normal to f by Fact 3.3.
Suppose |I|>4 . We have \delta(B_{I})=2+|I|/8 , and |B_{l}|=2+2|I|. |B_{I}| is12, 14, 16, 18, 20 for |I|=5,6,7,8,9 , respectively.
We also havef(10)=2+4/8, f(17)=2+5/8, f(29)=2+6/8 , andf(51)=2+7/8.
Suppose |I|=5. |U|\leq 12+10=22 , and \delta(U)\geq 2+5/8+1/8=2+
6/8 . Since |U|\leq 22<29, U is normal to f.
Suppose |I|=6. |U|\leq 14+12=28 , and \delta(U)\geq 2+6/8+1/8=2+
7/8>f(29) . Since |U|\leq 28<29, U is normal to f.

84
Suppose |I|=7. |U|\leq 16+14=30 , and \delta(U)\geq 2+7/8+1/8=2+1.
|U|\leq 30<51 , and f(51)=2+7/8<\delta(U), U is normal to f.
Suppose |I|=8. |U|\leq 18+16=34 , and \delta(U)\geq 2+8/8+1/8=2+
9/8 . Since |U|\leq 30<51 , and f(51)=2+7/8<\delta(U), U is normal to f.
Finally, suppose |I|=9. |U|\leq 20+18=38 , and \delta(U)\geq 2+9/8 . Since|U|\leq 38<51 , and f(51)=2+7/8<\delta(U), U is normal to f.
Now, consider the general case. We can assume that A\subseteq U , and U is

Sincesmooth over AS\cap U . Let I be the set of i such that B_{i}\subseteq U\cap H_{i}.

Let U_{I}=\otimes_{AS\cap U}\{H_{i}\cap U\}_{i\in l} , and j\not\in I.
Put U_{j}^{\prime/}=U_{I}\otimes_{AS\cap U}H_{j} . Then U_{I} and U_{j}^{\prime/} are normal to f by Subcase 1.

Put U_{j}=U_{I}\otimes_{AS\cap U}(H_{j}\cap U) . Then

|U_{j}|<|U_{j}"| and \delta(U_{j})\geq\delta(U_{j}") because\delta(H_{j}\cap U/AS\cap H_{j}\cap U)\geq\delta(H_{j}/AS\cap H_{j}) . Hence, U_{j} is normal to f . Also,

we have U_{I}<U_{j} . Therefore,

U= \bigotimes_{U_{I} \{U_{j}\}_{j\not\in I}
is normal to f by Fact 3.2.

\squareLemma 4.2. Let C_{2}=C_{1}\otimes_{c_{1}}c_{1} pc where c_{1} pc is a path of length 2. Then

C_{2}\in K_{f} and Ac<C_{2}.

Proof. C_{2}\in K_{f} because C_{1} and c_{1} pc belong to K_{f} and one point is always
closed in any structure.
By inspection, for any U\subseteq C_{1} with A\subseteq U and c_{1}\in U , we have \delta(U/A)\geq
1 . Therefore, for any U\subseteq C_{2} with A\subseteq U and c\in U , we have \delta(U/A)>1.
Therefore,\delta(U/Ac)=\delta(U/A)-1>0 . Hence, Ac<C_{2}.

\squareLemma 4.3. Let C_{3}=\otimes_{Ac}\{C_{11}, C_{12},C_{13},C_{14}\} where C_{1j} are isomorphic to
C_{1} overAc. Then C_{3}\in K_{f}, A<C_{3}, Ac<C_{3}.

5. MONODIMENSIONALITY

We can prove the following theorem as in [5].
Theorem 5.1. The generic structure M of K_{f} is monodimensional. There‐
fore, the automorphism group of M is a simple group.
Proof. Consider C_{3} from 4.3. We can assume a_{1} and a_{2} are not connected
and A=\{a_{1},a_{2}\}<M . So, we can embed C_{3} as a closed substructure of M.
We can assume that C_{3}<M . Isomorphic images of B over A in C_{3} are also
closed in M . Therefore, isomorphic images of b_{1} are in a same basic orbit

85
over A , say D . Hence, c\in c1(A,D) . Since cA<C_{3}, cA is closed in M andthus c\perp A.

By Fact 2.6, we have \{e\in M|e\perp A\}\subseteq c1(A,D) .
Now, let x\in M-A . Let X=c1(xA) . Consider the following structure U :Then X<X\otimes_{x}U . Embed this structure over X as a closed structure ofM.

So, we can assume XU<M . For each leaf y of U, xy<U . Therefore,
Ay<M . Hence, y\perp A . On the other hand, by the structure of U, x is in
the closure of the leaves of U . Thus, x\in c1(\{e\in M|e\perp A\}) . Therefore,

x\in c1(A,D)\square.

ACKNOWLEDGMENTS

The work is supported by JSPS KAKENHI Grant Number 17K05345,
and by the Research Institute for Mathematical Sciences, an International
Joint Usage/Research Center located in Kyoto University.
REFERENCES[1] J.T. Baldwin and K. Holland, Constructing \omega ‐stable structures: model completeness,
Ann. Pure Appl. Log. 125,159−172 (2004).
[2] J.T.
BaldwinandS.
Shelah,Randomness
andsemigenericity,
Trans. Am. Math. Soc. 349, 1359‐1376 (1997).
[3] J.T. Baldwin and N. Shi, Stable generic structures, Ann. Pure Appl. Log. 79,1−35(1996).

[4] R. Diestel, Graph Theory, Fourth Edition, Springer, New York (2010).
[5] D. Evans, Z. Ghadernezhad, and K. Tent, Simplicity of the automorphism groups ofsome Hrushovski constructions, Ann. Pure Appl. Logic 167, 22‐48 (2016).
[6] K. Holland, Model completeness of the new strongly minimal sets, J. Symb. Logic
64, 946‐962 (1999).
[7] E. Hrushovski, A stable \aleph_{0} ‐categorical pseudoplane, preprint (1988).
[8] E. Hrushovski, A new strongly minimal set, Ann. Pure Appl. Log. 62,147−166(1993).

86[9] K. Ikeda, H. Kikyo, Model complete generic structures, in the Proceedings of the
13th Asian Logic Conference, World Scientific, 114‐123 (2015).
[10] H. Kikyo, Model complete generic graphs I, RIMS Kokyuroku 1938, 15‐25 (2015).
[11] H. Kikyo, Balanced Zero‐Sum Sequences and Minimal Intrinsic Extensions, RIMS
Kokyuroku 2079, Balanced zero‐sum sequences and minimal intrinsic extensions(2018).

[12] H. Kikyo, Model Completeness of Generic Graphs in Rational Cases, Archive forMathematical Logic 57 (7‐8), 769‐794 (2018).
[13] H. Kikyo, Model completeness of the theory of Hrushovski’s pseudoplane associatedto 5/8, RIMS Kokyuroku 2084, 39‐47 (2018).
[14] H. Kikyo, S. Okabe, On automorphism groups of Hrushovski’s pseudoplanes in ra‐
tional cases, in preparation.
[15] F.O. Wagner, Relational structures and dimensions, in Automorphisms of first‐order
structures, Clarendon Press, Oxford, 153‐181 (1994).
[16] F.O. Wagner, Simple Theories, Kluwer, Dordrecht (2000).

Graduate School of System Informatics
Kobe University
1‐1 Rokkodai, Nada, Kobe 657‐8501
JAPANkikyo@kobe‐u.ac.jp