RELATIONAL QUOTIENTS
MIODRAG SOKIĆ
Abstract. Let K be a class of …nite relational structures. We de…ne
EK to be the class of …nite relational structures A such that A=E 2 K,
where E is an equivalence relation de…ned on the structure A. Adding
arbitrary linear orderings to structures from EK, we get the class OEK.
If we add linear orderings to structures from EK such that each Eequivalence class is an interval then we get the class CE[K ]. We provide
a list of Fraïssé classes among EK, OEK and CE[K ]. In addition to
this, we classify OEK and CE[K ] according to the Ramsey property.
We also conduct the same analysis after adding additional structure
to each equivalence class. As an application of these results, we give
a topological interpretation using the technique introduced in [11]. In
particular, we extend the lists of known extremely amenable groups and
universal minimal ‡ows.
1. Introduction
A standard mathematical approach is to take a structure A with an equivalence relation E and de…ne its quotient structure A=E. In this paper, we
present the opposite approach. We start with a relational structure B and
consider a relational structure A with relation E such that A=E = B.
Roughly speaking, we replace points by …nite equivalence classes. It appears that we get a class structures suitable for application of the technique
developed by Kechris-Pestov-Todorµcević (KPT), see [11].
We start with a class K of …nite relational structures in a given signature
L = fRi gi2I . We then de…ne the class EK of …nite relational structures in
the signature L [ fEg, E 2
= L, such that for all A 2 EK we have A=E 2 K.
The structure A=E will be de…ned in De…nition 1. We continue by adding
linear orderings to structures from EK. Adding arbitrary linear orderings
Date: April 25, 2012.
1991 Mathematics Subject Classi…cation. 03C15, 05D10, 54H20.
Key words and phrases. Ramsey property, Fraïssé class, Linear ordering.
1
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MIODRAG SOKIĆ
to structures from EK, we get the class OEK. We say that a linear ordering
de…ned on the structure A 2 EK, which has the equivalence relation
E, is convex if each E-equivalence class is an interval with respect to .
Let the class K be obtained by adding linear orderings to structures from
K in some way. The class K is a subclass of the class of all …nite order
expansions of K. We de…ne the class CE[K ] OEK of structures A such
that A=E 2 K .
Let L be a class of …nite relational structures in a given signature L0 =
fRj gj2J such that L \ L0 = ;. We de…ne the class LEK of …nite relational
structures in the signature L [ L0 . Structures from LEK are obtained by
putting a structure from L on each equivalence class of a structure from
EK. On distinct equivalence classes we may put di¤erent structures from
L. If we add arbitrary linear orderings to structures from LEK, we obtain
the class OLEK. Let the classes K and L be obtained by adding linear
orderings to structures from K and L respectively. Using K and L , we
de…ne the class C[L ]E[K ], which is obtained from LEK by adding linear
orderings to structures from LEK. If A 2 OLEK then by dropping relations
which interprete symbols from L0 we obtain a structure A from OEK. If
A 2 CE[K ], and the ordered structures de…ned on each equivalence class
with respect to L0 belongs to L , then A belong to the class C[L ]E[K ].
Fraïssé classes and structures are extensively studied in model theory; see
[6] and [10]. Fraïssé classes of graphs have been classi…ed in [12]. The class
of hypergraphs of a given type is also a Fraïssé class; see [11]. The list of
known Fraïssé classes includes …nite vector spaces over a given …nite …eld (
see [29]), …nite boolean algebras (see [11]), …nite metric spaces with rational
distances ( see [14]), and …nite ultrametric spaces ( see [13]). In particular,
we are interested in extending the list of known Fraïssé classes. In Section
3 of this paper, we extend this list with the following results.
Theorem 1. If K and K are Fraïssé classes then EK, OEK and CE[K ]
are Fraïssé classes.
We denote by EK, OEK and CE[K ] the Fraïssé limits of EK, OEK and
CE[K ] respectively.
Theorem 2. If K, L, K and L are Fraïssé classes then LEK and C[L ]E[K ]
are Fraïssé classes.
RELATIONAL QUOTIENTS
3
Note that OLEK is not always a Fraïssé class, see Section 3. If LEK,
OLEK and C[L ]E[K ] are Fraïssé classes then we denote their Fraïssé limits
by LEK, OLEK and C[L ]E[K ] respectively.
Let C and B be given structures. If C is isomorphic to B, we write
C = B, but if C is a substructure of B we write C B. Also, we use the
following notation
(B
A ) = fC : C
B; C = Ag:
A class of structures K has the Ramsey property (RP) if for every natural
number r and every A, B 2 K there is C 2 K such that for every coloring
there is B0 2 (C
B ) satisfying:
c : (C
A ) ! f1; ::; rg;
0
c (B
A ) = const:
If structures A, B and C satisfy the previous statement, we use the arrow
notation
C ! (B)A
r:
Section 4 is devoted to an analysis of the Ramsey property for the classes
OEK and CE[K ]. The list of Ramsey classes includes the class of linearly
ordered graphs (see [1] and [18]), the class of linearly ordered …nite metric
spaces (see [15]), and the class of …nite convexly ordered ultrametric spaces
(see [13]). The following results are generalizations of a result from [16] in
the sense that we examine general relational structures instead of partial
orderings.
Theorem 3. If K is a Ramsey class, then CE[K ] is a Ramsey class.
Theorem 4. If K and L are Ramsey classes, then C[L ]E[K ] is a Ramsey
class.
We say that a structure A is rigid if its group of automorphisms Aut(A)
is trivial. Ramsey classes of rigid structures are of particular interest.
Theorem 5. If K is a Ramsey class of rigid structures, then OEK is a
Ramsey class.
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MIODRAG SOKIĆ
We obtain a similar result for the class OLEK under additional requirements to the class L; see Section 4.
In Section 6 we present a topological application of our results. Let G
be a topological group. A G-‡ow X is a continuous action G X ! X on
a compact Hausdor¤ space X. A G-‡ow X is called minimal if each of its
orbits is dense, i.e. for all x 2 X , X = G x. Among all minimal G-‡ows
there is a largest one called the universal minimal G-‡ow, see [2]. If every
G-‡ow X has a …xed point x 2 X, gx = x for all g 2 G, then we say that
G is an extremely amenable group. The list of extremely amenable groups
includes the group of increasing homeomorphisms of the unit interval (see
[21]), the unitary group of in…nite dimensional separable Hilbert space (see
[8]), the group of isometries of Urysohn space (see [22]), the pathological
groups (see [9]), and the group of automorphisms of the rationals (see [21]).
The technique from [11] helps us to extend this list and to calculate some
universal minimal ‡ows. In Section 6 we present the calculation of the
universal minimal ‡ow for the group Aut(EK), as well as the following.
Theorem 6. If K is a Ramsey class which satis…es the ordering property
with respect to K then, Aut(CE[K ]) is an extremely amenable group.
Theorem 7. If K and L are Ramsey classes, then Aut(C[L ]E[K ]) is
an extremely amenable group.
Section 7 is devoted to applications of our results to the case where K is
the class of …nite posets. Then EK is the class of …nite quasi ordered sets.
We use the connection between the class of …nite topological spaces and
the class of …nite quasi ordered sets, so our results can be viewed as results
about linearly ordered …nite topological spaces. Section 8 is dedicated to
the application of our results to the case where K is the class of …nite
metric spaces with rational distances, so EK represents the class of …nite
pseudometric spaces with rational distances. In Sections 9, 10 and 11 we
give applications our results to the class of ultrametric spaces, the class
of graphs and the class of chains respectively. In section 12 we give some
modi…cation on de…nitions of our structures.
Acknowledgement 1. The author is grateful to the referee for pointing
out many errors and suggesting improvements.
RELATIONAL QUOTIENTS
5
2. Preliminaries
A relational signature is a collection of distinct relational symbols L =
fRi gi2I . Each relational symbol Ri has an assigned non-zero natural number
ni , called arity.
A structure in a relational signature L = fRi gi2I with arity fni gi2I is a
set A together with relations fRiA gi2I on the set A such that for all i 2 I:
RiA Ani . RiA is called the interpretation of the relational symbol Ri in the
structure A = (A; fRiA gi2I ).
Let B = (B; fRiB gi2I ) be a structure in signature L. The map f : A ! B
is called a morphism if for all i 2 I and all x1 ,...,xni 2 A the following holds:
RiA (x1 ; :::; xni ) () RiB (f (x1 ); :::; f (xni )):
In this case we write f : A ! B. An injective morphism is called an
embedding, while a bijective morphism is called an isomorphism. If there is
an embedding of the structure A into the structure B, we write A ,! B,
while isomorphic structures are denoted by A = B. A structure A is
a substructure of the structure B if A
B, and for all i 2 I we have
ni
B
A
Ri = Ri \ A , or in symbols A B.
In this paper we consider only classes of …nite structures closed under
isomorphisms.
A class of …nite structures K has the:
(1) hereditary property (HP) if for A 2 K and B A we have B 2 K;
(2) joint embedding property (JEP) if for all structures A; B 2 K there
is C 2 K such that A ,! C and B ,! C;
(3) amalgamation property (AP) if for all structure A; B; C 2 K and
all embeddings iB : A ! B, iC : A ! C there are D 2 K and
embeddings jB : B ! D, jC : C ! D such that jB iB = jC iC ;
(4) strong joint embedding property (SJEP) if for all structures A and
B from K de…ned on sets A and B respectively, there are C 2 K and
embeddings iA : A ! C, iB : B ! C such that iA (A) \ iB (B) = ;;
(5) strong amalgamation property (SAP) if for all structures A; B; C 2K
de…ned on sets A; B; C respectively, and all embeddings iB : A ! B,
iC : A ! C there are D 2 K and embeddings jB : B ! D,
jC : C ! D such that jB (B) \ jC (C) = jB iB (A) = jC iC (A).
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MIODRAG SOKIĆ
It should be clear that SJEP and SAP imply JEP and AP respectively.
For our analysis the properties HP, JEP and AP are of main importance,
and properties SJEP and SAP appear naturally.
Consider two relational signatures LI = fRi gi2I and LJ = fRi gi2J such
that I
J. Assume that the structures AI = (A; fRiAI gi2I0 ) in signature
LI , and AJ = (A; fRiAJ gi2J ) in signature J, are given on the same set A.
If for all i 2 I we have RiAI = RiAJ then we say that the structure AJ is
an expansion of the structure AI or that the structure AI is a reduct of
J
the structure AJ . We write AI = AJ jLI and AJ = (AI ; fRA
i gi2JnI ). For a
class KJ of structures in signature LJ , we denote the class of their reducts
by: KJ jLI = fAjLI : A 2 Kg = KI , and we say that KJ is an expansion of
the class KI or that KI is a reduct of the class KJ .
Let LJ = LI [f<g be a signature such that < is a binary relational symbol
satisfying <2
= LI . Assume that KI and KJ are classes of …nite structures
in signatures LI and LJ respectively such that symbol < is interpreted as
linear ordering in each structure from KJ . If KJ jLI = KI then:
(1) We say that the class KJ is an ordered class.
(2) If for every AI ; BI 2 KI , every linear ordering <AI satisfying (AI ; <AI
) 2 KJ , and every embedding : AI ! BI there is a linear ordering
<BI such that (BI ; <BI ) 2 KJ and is also an embedding from
(AI ; <AI ) into (BJ ; <BJ ), then we say that KJ is a reasonable expansion of KI .
(3) If for every AI ; BI 2 KI and all linear orderings <AI and <BI satisfying (AI ; <AI ) 2 KJ and (BI ; <BI ) 2 KJ we have (AI ; <AI ) ,!
(BI ; <BI ), then we say that KJ has the ordering property (OP) with
respect to KI .
De…nition 1. Let K be a class of …nite relational structures in the signature
L = fRi gi2I , and let E be a binary relational symbol such that E 2
= L. Then
EK is the class of relational structures of the form A = (A; fRiA gi2I ; E A )
in the signature L [ fEg such that:
(1) E A is an equivalence relation on the set A,
(2) For all i 2 I and all x1 ; :::; xni ; y1 ; :::; yni 2 A with xj E A yj , 1
ni , we have
RiA (x1 ; :::; xni ) , RiA (y1 ; :::; yni );
j
RELATIONAL QUOTIENTS
7
A=E
(3) A=E = (A=E; fRi gi2I )2K , where A=E = f[a]E A : a 2 Ag, is the
set of equivalence classes of the form [a]E A = fx 2 A : aE A xg, where
A=E
the relations fRi gi2I are well-de…ned on the set A=E A according
to the condition 2 with
A=E
Ri
([a1 ]E A ; :::; [ani ]E A ) , RiA (x1 ; :::; xni );
where for all j we have aj E A xj .
Note that each element of the class K can be treated as an element of
the class EK, where each equivalence class has exactly one element. Let
A = (A; fRiA gi2I ; E A ) and B = (B; fRiB gi2I ; E B ) be two structures from
the class EK, and let f : A ! B be a morphism. Then for all x, y 2 A we
have: xE A y , f (x)E B f (y). Consequently, each morphism of structures
from EK produces the injective map from the set A=E into the set B=E,
and moreover this induces embedding of the structures from the class K.
We denote the map in the class K induced by f with fE : A=E ! B=E.
Approach from [11] needs examination of ordered classes. Naturally, we
add linear orderings to structures from EK.
De…nition 2. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let E and be binary relational symbols such that E 2
= L and
2
= L. Then OEK is the class of structures of the form (A; fRiA gi2I ; E A ; A
) such that (A; fRiA gi2I ; E A ) 2 EK and A is a linear ordering on A.
Let A = (A; fRiA gi2I ; E A ) be a structure from EK. A linear ordering
on the set A which satis…es for all x, y, z 2 A:
A
x A y A z; xE A z ) xE A yE A z;
is called convex with respect to E A .
Let K be an order expansion of the class K. Structures from K have the
form (A; fRi gi2I ; ) such that (A; fRi gi2I ) 2 K and is a linear ordering
on the set A, and =
6 Ri for i 2 I.
De…nition 3. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let E, and be binary relational symbols such that E 2
= L,
2
= L and 2
= L. Let K be a class of …nite relational structures in a
signature L [ f g such that K is an ordered expansion of K and K jL =
K. Then CE[K ] is the class of structures A = (A; fRiA gi2I ; E A ; A ) 2
OEKsuch that:
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MIODRAG SOKIĆ
(1) A is convex with respect to E A ,
A=E
(2) A=E = (A=E; fRi gi2I ; A=E ) 2 K , where linear ordering
on A=E is well-de…ned according to 1. with:
[a]E A
A=E
[b]E A , ([a]E A = [b]E A ) or ([a]E A 6= [b]E A ; a
A
A=E
b);
for all a, b 2 A.
For the rest of this section we …x the relational signatures LI = fRi gi2I
and LJ = fRi gi2J with arity fni gi2I and fnj gj2J respectively such that
LI \ LJ = ;. We also assume that E and are binary symbols which do
not belong to LI [ LJ . Let K be a class of …nite relational structures in LI ,
and let L be a class of …nite relational structures in LJ .
De…nition 4. LEK is the class of relational structures of the form A =
(A; fRiA gi2I ; E A ; fRjA gj2J ) in the signature LI [ fEg [ LJ such that:
(1) (A; fRiA gi2I ; E A ) 2 EK,
(2) For all j 2 J and all x1 ; :::; xnj 2 A we have
RjA (x1 ; :::; xnj ) ) [x1 ]E A =
(3) For all a 2 A we have
= [xnj ]E A ;
([a]E A ; fRjA \ ([a]E A )nj gi2J ) 2 L:
We will represent the structures from the class LEK in abbreviated notaA=E
tion. Let C be the structure with underlying set C such that C =(A=E; fRi gi2I ) 2
K. Let Dc be the structure from L such that Dc = (c; fRjA \(cnj gj2J ). Then
we denote structure A from the previous De…nition by C; (Dc )c2C .
Note that each element of the class LEK is obtained by transforming
the points of some structure from K into equivalence classes which have
their own structure from the class L. We recall that each element of
the class K can be treated as an element of the class EK, where each
equivalence class has exactly one element. This is not the case for the
class LEK in general since points may carry additional structure. Let
A = (A; fRiA gi2I ; E A ; fRjA gi2J ) and B = (B; fRiB gi2I ; E B ; fRjB gi2J ) be
two structures from the class LEK, and let f : A ! B be a morphism.
Then we have the induced morphism of reducts, which we also denote by f ,
f : Aj(LI [ fEg)! Bj(LI [ fEg), and a sequence of morphisms ffa ga2A of
structures in L. Each fa is a morphism from the substructure of A induced
RELATIONAL QUOTIENTS
9
on [a]E A into the substructure of B induced on [f (a)]E B . As in the case of
the class EK, the map f produces an injective map from the set A=E into
the set B=E, and an embedding of the structures from the class K, which
we denote by fE : A=E ! B=E.
If we add arbitary linear orderings to structures from LEK, then we obtain
the following class.
De…nition 5. OLEK is the class of structures of the form
(A; fRiA gi2I ; E A ; fRjA gi2J ;
A
)
such that A = (A; fRiA gi2I ; E A ; fRjA gi2J ) 2 LEK and
ing on A.
A
is a linear order-
If in the previous de…nition we denote the structure A by C; (Dc )c2C ,
then the structure obtained by adding the linear ordering A we denote by
C; (Dc )c2C ; A .
Let K and L be order expansions of the classes K and L respectively.
Structures from K and L have the form (A; fRi gi2I ; A ) and (B; fRj gj2J ;
) such that (A; fRi gi2I ) 2 K, (B; fRj gj2J ) 2 L and A and B are
linear orderings on the sets A and B respectively. We assume that 2
=
LI [ LJ [ Ri [ fE; g.
De…nition 6. C[L ]E[K ] is the class of structures
A = (A; fRiA gi2I ; E A ; fRjA gi2J ;
A
) 2 OLEK
such that:
(1) (A; fRiA gi2I ; E A ; A ) 2 CE[K ],
(2) ([a]E A ; fRjA \ ([a]E A )nj gi2J ; A \([a]E A )2 ) 2 L for every a 2 A.
Let in the previous De…nition structure (A; fRiA gi2I ; E A ; fRjA gi2J ) be denoted by C; (Dc )c2C . The linear ordering A induces a linear ordering
C
on C and linear orderings Dc for each structure Dc , c 2 C, such that
(C; C ) 2 K and (Dc ; Dc ) 2 L for all Dc 2 C. Then we denoteE the
structure A from the previous de…nition by (C; C ); ((Dc ; Dc ))c2C .
Note that in the class C[L ]E[K ], the linear ordering restricted to an
equivalence class can depend on the structure, but in the class CE[K ] this
is not the case.
B
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MIODRAG SOKIĆ
3. Fraïssé classes
A structure A is called ultrahomogenous if every isomorphism between
…nite substructures S and T of A can be extended to an automorphism
of A. A countable in…nite structure which is ultrahomogenous and has
the property that every …nitely generated substructure is …nite is called a
Fraïssé structure. Note that every relational structure has the property that
every …nitely generated substructure is …nite. We say that a class of …nite
structures is countable if the maximal set of its non-isomorphic structures
is countable. Given a signature L, a Fraïssé class in L is a countable class
of …nite structures in L which contains structures of arbitrary large …nite
cardinality and has HP, JEP, and AP. The age of a structure A is the class
Age(A) of all …nitely generated substructures that can be embedded into
A. The following two results establish a connection between the Fraïssé
classes and the Fraïssé structures, see [11, 6, 10].
Theorem 8. Let A be a Fraïssé structure. Then Age(A) is a Fraïssé class.
Theorem 9. Let K be a Fraïssé class. Then there is a unique, up to isomorphism, countable structure A such that A is a Fraïssé structure and
K =Age(A).
The structure from the Theorem 14 is called the Fraïssé limit of K, we
write A =F lim(K).
We examine classes EK, CE[K ] and OEK in order to recognize Fraïssé
classes among them.
Lemma 1. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . If EK is the class of …nite structures given by De…nition 1,
we have:
(i) If class K has HP then EK has HP.
(ii) If class K has JEP then EK has SJEP.
(iii) If class K has AP then EK has SAP.
Proof. (i) If A and B are such that A B and B 2EK, then A=E B=E
and B=E 2K. HP for the class K implies A=E 2K, so A 2EK.
(ii) Let A = (A; fRiA gi2I ; E A ) and B = (B; fRiB gi2I ; E B ) be two structures from the class EK. Then A=E, B=E 2K and, JEP of the class K
provides a structure C = (C; fRiC gi2I ) 2 K and embeddings iA : A=E ! C,
iB : B=E ! C. Using the structure C, we de…ne D = (D; fRiD gi2I ; E D )2EK
RELATIONAL QUOTIENTS
11
replacing points from C with E D -equivalence classes such that D=E = C.
For c 2 C we denote E D -equivalence class given with c by c^. For c 2 C we
de…ne sets Ac , Bc , and Cc as follows:
(1) If there are a 2 A, b 2 B such that iA ([a]E A ) = c = iB ([b]E B ), then
c^ = Ac [ Bc , Ac \ Bc = ;, jAc j = j[a]E A j, jBc j = j[b]E B j, Cc = ;.
(2) If there is a 2 A such that iA ([a]E A ) = c, but there is no b 2 B such
that c = iB ([b]E B ), then c^ = Ac , jAc j = j[a]E A j, Bc = ; = Cc .
(3) If there is b 2 B such that c = iB ([b]E B ), but there is no a 2 A such
that iA ([a]E A ) = c, then c^ = Bc , jBc j = j[b]E B j, Ac = ; = Cc .
(4) If there is no a 2 A such that iA ([a]E A ) = c, and there is no b 2 B
such that c = iB ([b]E B ) , then c^ = Cc , jCc j = 1, Ac = ; = Bc .
The set D is the disjoint union:
D=
[
c2C
fRiD gi2I
(Ac [ Bc [ Cc ):
are de…ned in the only possible way such that D=E = C,
Relations
i.e. for d1 ,..., dni 2 D we have
RiD (d1 ; :::; dni ) () RiC ([d1 ]E D ; :::; [dni ]E D ):
We de…ne functions ^{A : A ! D and ^{B : A ! D such that:
(1) For c 2 C, a 2 A satisfying iA ([a]E A ) = c, the map ^{A [a]E A ! Ac
is a bijection.
(2) For c 2 C, b 2 B satisfying iB ([b]E B ) = c, the map ^{B [b]E B ! Bc
is a bijection.
Clearly, ^{A (A) \ ^{B (B) = ;, and ^{A and ^{B are structure embeddings, so
SJEP holds.
(iii) We consider the following structures
A = (A; fRiA gi2I ; E A ); B = (B; fRiB gi2I ; E B ); C = (C; fRiC gi2I ; E C )
from EK and embeddings f : A ! B and g : A ! C. Then we notice
structures A=E, B=E and C=E with induced embeddings fE : A=E ! B=E
and gE : A=E ! C=E. Since the class K satis…es AP, there is a structure
D = (D; fRiD gi2I ) 2K, and embeddings f^ : B=E ! D and g^E : C=E ! D
such that f^E fE = g^E gE . Using the structure D, we de…ne a structure
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MIODRAG SOKIĆ
F = (F; fRiF gi2I ; E F ) from EK by replacing each point d 2 D with the
^ Each d^ is a disjoint union:
E F -equivalence class d.
d^ = Ad [ Bd [ Cd [ Dd :
For d 2 D we de…ne sets Ad , Bd , Cd , Dd as follows:
(1) If there is a 2 A such that f^E fE ([a]E A ) = d = g^E gE ([a]E A ), then
jAd j = j[a]E A j, jBd j = j[f (a)]E B nf ([a]E A )j, jCd j = j[g(a)]E C ng([a]E A )j,
Dd = ;.
(2) If there are b 2 B, c 2 C such that f^E ([b]E B ) = d = g^E ([c]E C ), and
there is no a 2 A such that f^E fE ([a]E A ) = d = g^E gE ([a]E A ),
then jBd j = j[b]E B j, jCd j = j[c]E C j, Ad = ; = Dd .
(3) If there is b 2 B such that f^E ([b]E B ) = d, but there is no c 2 C such
that d = g^E ([c]E C ), then jBd j = j[b]E B j, Ad = Cd = Dd = ;.
(4) If there is c 2 C such that g^E ([c]E C ) = d, but there is no b 2 B such
that f^E ([b]E B ) = d, then jCd j = j[c]E C j, Ad = Bd = Dd = ;.
(5) If there is no b 2 B such that f^E ([b]E B ) = d, and there is no c 2 C
such that d = g^E ([c]E C ), then jDd j = 1, Ad = Bd = Cd = ;.
The set F is the disjoint union:
[
d2D
(Ad [ Bd [ Cd [ Dd ):
The relations fRiF gi2I are de…ned in the only possible way such that F=E =
D, i.e. for f1 ,..., fni 2 F we have
RiF (d1 ; :::; dni ) () RiD ([f1 ]E F ; :::; [fni ]E F ):
We de…ne an embedding f : B ! F, and we de…ne an embedding g :
C ! F. To de…ne f it is enough to de…ne a restriction of f on [b]E B for all
b 2 B:
(1) If there is a 2 A such that f^E ([a]E A ) = [b]E B , then functions f
([b]E B nf ([a]E A ) ! Bd and f f ([a]E A ) ! Ad are bijections where
d 2 D is such that f^E ([b]E B ) = d.
(2) If there is no a 2 A such that f^E ([a]E A ) = [b]E B , then the restriction
f [b]E B ! Bd is bijection where d 2 D is such that f^E ([b]E B ) = d.
To de…ne g we consider all c 2 C:
RELATIONAL QUOTIENTS
13
(1) If there is a 2 A such that g^E ([a]E A ) = [c]E C , then there is b 2 B
such that f^E ([a]E A ) = [b]E B . Map g [c]E C ng([a]E A ) ! Cd is a
bijection, and the map g g([a]E A ) ! Ad is a bijection such that
g g [a]E A = f f [a]E A where g^E ([c]E C ) = d.
(2) If there is no a 2 A such that g^E ([a]E A ) = [c]E C , then restriction
g [c]E C ! Cd is bijection where d 2 D is such that g^E ([c]E C ) = d.
It is straightforward to check that f f = g g. The above construction
implies that f (B) \ g(C) = f f (A) = g g(A) and clearly f and g are
structure embeddings, so SAP is veri…ed.
Lemma 2. Let K be a class of …nite relational structures in a signature
LI = fRi gi2I . Let L be a class of …nite relational structures in a signature
LJ = fRj gj2J . If ELK is the class of …nite structures given by De…nition
4, then we have:
(i) If K and L have HP then LEK has HP.
(ii) If K and L have JEP then LEK has JEP.
(iii) If one of the classes K and L have SJEP and the other has JEP then
LEK has SJEP.
(iv) If K and L have AP then LEK has AP.
(v) If K has AP and L has SAP then LEK has SAP.
The proof of the Lemma 2 is similar to the proof of the Lemma 1, so we
omit it. Note that the class EK always has SAP if K has AP, but the class
LEK will not have always SAP. A similar conclusion holds for SJEP. The
following result will help us to examine JEP and AP for the class OEK and
OLEK.
Lemma 3. [11]Let L be a relational signature and let <2
= L be a binary
relational symbol. Let K be a class of structures in L and let
OK = f(A; <) : A 2 K and < is a linear ordering on underlying set of Ag:
(i) OK satis…es SJEP() K satis…es SJEP.
(ii) OK satis…es SAP() OK satis…es AP() K satis…es SAP.
The Previous two lemmas give the following result.
Lemma 4. Let K be a class of …nite relational structures in a signature L.
If EK and OEK are de…ned as in De…nitions 1 and 2, we have:
(i) If K satis…es HP then OEK satis…es HP.
(ii) K satis…es JEP then OEK satis…es SJEP.
14
MIODRAG SOKIĆ
(iii) K has AP then OEK satis…es SAP.
Lemma 5. Let K be a class of …nite relational structures in a signature
LI = fRi gi2I . Let L be a class of …nite relational structures in a signature
LJ = fRj gj2J . If ELK and OLEK are de…ned as in De…nitions 4 and 5,
we have:
(i) If K and L have HP then OLEK satis…es HP.
(ii) If one of the classes K and L have SJEP and the other has JEP then
OLEK has SJEP.
(iii) If a class K has AP and class L has SAP then OLEK has SAP.
Lemma 6. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let
be a binary relational symbols such that 2
= L and let
K be a class of …nite relational structures in a signature L [ f g such that
K is an ordered expansion of K and K jL = K. If CE[K ] is the class of
structures given by De…nition 3, we have:
(i) If K satis…es HP then CE[K ] satis…es HP.
(ii) If K satis…es JEP then CE[K ] satis…es SJEP.
(iii) If K satis…es AP then CE[K ] satis…es SAP.
Proof. (i) If A and B are such that A B and B 2CE[K ], then A=E B=E
and B=E 2K . HP for the class K implies A=E 2K , so A 2CE[K ].
(ii) Let A = (A; fRiA gi2I ; E A ; A ) and B = (B; fRiB gi2I ; E B ; B ) be
A=E
structures from the class CE[K ]. Then A=E = (A=E; fRi gi2I ; A=E )
B=E
and B=E = (B=E; fRi gi2I ; B=E ) belong to the class K . JEP of the
class K provides a structure C = (C; fRiC gi2I ; C ) 2 K and embeddings iA : A=E ! C, iB : B=E ! C. Using the structure C we de…ne
D = (D; fRiD gi2I ; E D ; D )2CE[K ], replacing points from C with E D equivalence classes such that D=E = C. To each c 2 C we assign a class c^
from D.
Note that since K jL = K and K has JEP, we have that K has JEP.
Moreover (A; fRiA gi2I ; E A ) and (B; fRiB gi2I ; E B ) belong to EK. The emA=E
beddings iA and iB induce embeddings of structures (A=E; fRi gi2I ) and
B=E
(B=E; fRi gi2I ) into (C; fRiC gi2I ). In the same way as in the proof of
Lemma 1 part (ii) we de…ne:
(1) Sets Ac ; Bc ; Cc for all c 2 C:
(2) A structure (D; fRiD gi2I ; E D ) on D = [c2C (Ac [ Bc [ Cc ).
RELATIONAL QUOTIENTS
15
(3) Embeddings
^{A : (A; fRiA gi2I ; E A )!(D; fRiD gi2I ; E D );
^{B : (B; fRiB gi2I ; E B ) ! (D; fRiD gi2I ; E D )
such that ^{A (A) \ ^{B (B) = ;.
The linear ordering D on D is de…ned such that:
(1) Linear ordering D is convex with respect to E D and
i.e. for d1 , d2 2 D we have
[d1 ]E D 6= [d2 ]E D =) (d1
D
d2 , [d1 ]E D
D=E
D=E
=
C
,
[d2 ]E D ):
(2) For all c 2 C, each of the non empty sets Ac , Bc , Cc is convex with
respect to D .
(3) For all a 2 A, b 2 B the maps ^{A [a]E A and ^{B [b]E B are order
preserving.
This completes the de…nition of the structure D. Now, it is straightforward to see that ^{A and ^{B are embedding of structures A and B into D
such that ^{A (A) \ ^{B (B) = ;, so SJEP of the class CE[K ] is veri…ed.
(iii) Let A = (A; fRiA gi2I ; E A ; A ), B = (B; fRiB gi2I ; E B ; B ) and C =
(C; fRiC gi2I ; E C ; C ) be structures from CE[K ] and let f : A ! B and
A=E
g : A ! C be embeddings. We consider A=E = (A=E; fRi gi2I ; A=E ),
C=E
B=E
B=E = (B=E; fRi gi2I ; B=E ) and C=E = (C=E; fRi gi2I ; C=E ) from
K , and induced embeddings fE : A=E ! B=E and gE : A=E ! C=E.
AP of the class K implies existence of a structure D = (D; fRiD gi2I ; D
) 2 K and embeddings f^E : B=E ! D and g^E : C=E ! D such that
f^E fE = g^E gE . Replacing each point d 2 D with a set d^ we construct a
structure F = (F; fRiF gi2I ; E F ; F ) 2 CE[K ] and embeddings fE : B ! F
and gE : C ! F such that fE fE = gE gE .
Following the steps of the proof of Lemma 1 part (iii) we de…ne:
(1) Sets Ad ; Bd ; Cd :Dd for all d 2 D.
(2) A structure (F; fRiF gi2I ; E F ) on F = [d2D (Ad [ Bd [ Cd [ Dd ).
(3) Embeddings
f : (B; fRiB gi2I ; E B )!(F; fRiF gi2I ; E F );
g : (C; fRiC gi2I ; E C ) ! (F; fRiF gi2I ; E F )
16
MIODRAG SOKIĆ
such that f f = g g and f (B) \ g(C) = f f (A) = g g(A).
The linear ordering F is de…ned to be convex with respect to E F , and
such that for all d 2 D we have:
(1) Suppose there is a 2 A such that f^E fE ([a]E A ) = d = g^E gE ([a]E A ).
Consider linear orderings Ad ; Bd ; Cd on Ad ; Bd [ Ad ; Cd [ Ad respectively such that:
x
Ad
y , g 1 (x)
x
Bd
x
Cd
C
g 1 (y) , f
1
(x)
B
f
1
(y) for x; y 2 Ad ;
(x)
B
f
y , g 1 (x)
C
g 1 (y) for x; y 2 Cd [ Ad ;
y,f
1
1
(y) for x; y 2 Bd [ Ad ;
This means that restriction of the linear orderings Bd and Cd is
a linear ordering Ad . Therefore we can take F restricted to d^ to
be an arbitrary linear extension of the partial ordering generated by
Bd and
Cd .
(2) Suppose there are b 2 B, c 2 C such that f^E ([b]E B ) = d = g^E ([c]E C )
and there is no a 2 A such that f^E fE ([a]E A ) = d = g^E gE ([a]E A ).
Consider linear orderings Bd ; Cd on Bd ; Cd respectively such that:
x
Bd
x
Cd
(x)
B
f
y , g 1 (x)
C
g 1 (y) for x; y 2 Cd :
y,f
1
1
(y) for x; y 2 Bd ;
Then we can assume that the restriction of F to d^ is an arbitrary
linear extension of the partial ordering generated with Bd and Cd .
(3) Suppose there is b 2 B such that f^E ([b]E B ) = d, but there is no
c 2 C such that d = g^E ([c]E C ). Then restriction of F on d^ is such
that:
x F y , f 1 (x) B f 1 (y) for x; y 2 Bd :
(4) Suppose there is c 2 C such that g^E ([c]E C ) = d, but there is no
b 2 B such that f^E ([b]E B ) = d . Then the restriction of F on d^ is
such that:
x
F
y , g 1 (x)
C
g 1 (y) for x; y 2 Cd :
RELATIONAL QUOTIENTS
17
(5) Suppose there is no b 2 B such that f^E ([b]E B ) = d and there is no
c 2 C such that d = g^E ([c]E C ). Then there is only one choice for F
^ = 1.
restricted to d^ since jdj
It is straightforward to see that f and g are embedding such that fE fE =
gE gE and fE fE (A) = gE gE (A) = fE (B) \ gE (C), so SAP is veri…ed
for the class CE[K ].
The following Lemma is proved by arguments similar to Lemma 6, so we
skip its proof.
Lemma 7. Let K be a class of …nite relational structures in a signature
LI = fRi gi2I . Let L be a class of …nite relational structures in a signature
LJ = fRj gj2J . Let K and L be ordered expansion of K and L respectively.
If C[L ]E[K ] is the class of structures given by De…nition 6, we have:
(i) If K and L satis…es HP then C[L ]E[K ] satis…es HP.
(ii) If K and L satis…es JEP then C[L ]E[K ] satis…es JEP.
(iii) If one of the classes K and L have SJEP and the other has JEP
then C[L ]E[K ] satis…es SJEP.
(iv) If K and L satis…es AP then C[L ]E[K ] satis…es AP.
(v) If K satis…es AP and L satis…es SAP then C[L ]E[K ] satis…es SAP.
Let I be a countable index set, and let K be a class of …nite relational
structures in the signature L = fRi gi2I . If K is an order expansion of the
class K, then K is also countable. Moreover the classes EK , OEK, and
CE[K ] as de…ned in De…nitions 1, 2 and 3 respectively are also countable.
Therefore, for Fraïssé classes K and K with the property that K = K jL,
we have a list of Fraïssé classes
and their Fraïssé limits
EK; OEK; CE[K ];
EK =F lim(EK); OEK =F lim(OEK); CE[K ] = F lim(CE[K ]):
It is straightforward to see that OEK is a reasonable expansion of EK and
that CE[K ] is a reasonable expansion of EK if K is a reasonable expansion
of K.
Let K be a class of …nite relational structures in a signature LI = fRi gi2I .
Let L be a class of …nite relational structures in a signature LJ = fRj gj2J .
We assume that LI \LJ = ;. Let K and L be ordered expansions of K and
18
MIODRAG SOKIĆ
L respectively. If K and L are countable then K and L are countable, and
moreover the classes LEK , OLEK, and C[L ]E[K ] as de…ned in De…nitions
4, 5 and 6 respectively are also countable. Therefore, for Fraïssé classes K
and L, the class LEK is also Fraïssé with limit
LEK =F lim(LEK):
If K and L are Fraïssé classes such that one of them has SJEP and L has
SAP then OLEK is a Fraïssé class with limit
OLEK =F lim(OLEK):
If K and L are Fraïssé classes then C[L ]E[K ] is a Fraïssé class with limit
C[L ]E[K ]=F lim(C[L ]E[K ]):
It is straightforward to see that OLEK is a reasonable expansion of LEK and
that C[L ]E[K ] is a reasonable expansion of EK if K and L are reasonable
expansion of K and L respectively.
4. Ramsey property
A structure is rigid if the group of its automorphisms is trivial.
Lemma 8. [11]Let K be a class of …nite rigid structures. If K has HP,
JEP, and RP then K has AP.
The following is the well-known product Ramsey theorem, see [7].
Theorem 10. [7]Let l and r be natural numbers, and let (ai )li=1 and (bi )li=1
be two sequences of natural numbers satisfying ai bi for all i l. There
is a natural number c such that for all sets (Ci )li=1 with jCi j
c and any
coloring
p : fA1
A2
:::
Al : (8i
l)[Ai
Ci ; jAi j = ai ]g ! f1; :::; rg;
there is a sequence of sets (Bi )li=1 satisfying: Bi
and
p fA1
A2
:::
Al : (8i
l)[Ai
Ci , jBi j = bi for all i
Bi ; jAi j = ai ]g = const:
l
RELATIONAL QUOTIENTS
19
Using the Erdös-Rado arrow notation, for a c that satis…es the conclusion
of the previous theorem we write:
1 ;:::;al )
c ! (b1 ; :::; bl )(a
:
r
Theorem 11. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let
be a binary relational symbol such that 2
= L, and let
K be a class of …nite relational structures in a signature L [ f g such that
K is an ordered expansion of K and K jL = K. If CE[K ] is the class of
structures given by De…nition 3, then K satis…es RP i¤ CE[K ] satis…es
RP.
Proof. ()) We suppose that K satis…es RP in order to verify RP for the
class CE[K ]. Let A = (A; fRiA gi2I ; E A ; A ) and B = (B; fRiB gi2I ; E B ; B )
be two structures from the class CE[K ] such that (B
A ) 6= ;, and let r be a
A=E
…xed natural number. The structures A=E = (A=E; fRi gi2I ; A=E ) and
B=E
B=E = (B=E; fRi gi2I ; B=E ) belong to the Ramsey class K , so there is
a structure C = (C; fRiC gi2I ; C ) 2 K such that:
:
C ! (B=E)A=E
r
We de…ne the structure D = (D; fRiD gi2I ; E D ; D ) 2 CE[K ] such that
D=E
D=E = (D=E; fRi gi2I ; D=E ) = C. Each point from the set C will
be replaced with one E D -equivalence class, and the relations fRiD gi2I are
de…ned such that for all i 2 I, j ni , [xj ]E D = cj :
RiD (x1 ; :::; xni ) , RiC (c1 ; :::; cni ):
The linear ordering D is de…ned so that each E D -equivalence class is an
interval with respect to D , and for all [y1 ]E D = c1 , [y2 ]E D = c2 , c1 6= c2 :
y1
D
y2 , c1
C
c2:
All E D -equivalence classes will have the same number of elements and this
number is determined recursively. Let b = jBj and list all embeddings
A=E ,! C as fPj gpj=1 . We may assume that the E A -equivalence classes
(Ai )i l are linearly ordered with A=E in a chain :A1 A=E A2 A=E ::: A=E
Al , and let (ai )li=1 be a sequence such that ai = jAi j. We take b0 = b and
de…ne a sequence (bi )pi=0 recursively using Theorem 10:
20
MIODRAG SOKIĆ
1 ;:::;al )
:
bi+1 ! (bi ; :::; bi )(a
| {z } r
l times
All E -equivalence classes have the size bp . We claim that D ! (B)A
r . To
see this, consider a coloring:
D
p : (D
A ) ! f1; ::; rg:
Note that each embedding of the structure A into the structure D must
be placed on the collection of E D -equivalence classes given by one of the
(Pj )pj=1 . The choice of the number bp tell us that there is a Dp 1 2 CE[K ]
such that:
D
(1) Dp 1 = (Dp 1 ; fRi p 1 gi2I ; E Dp 1 ; Dp 1 ) 2 CE[K ]; Dp 1 D,
(2) all E Dp 1 -equivalence classes have the size bp 1 ,
(3) all embedding of A into D placed on the equivalence classes given
by Pp are monochromatic.
p 1
In general, there is a sequence of structures (Dj )j=0
from CE[K ] such that
Di Di+1 for all 0 i < p, equivalence classes of Di have the same size bi
and all embeddings of A into Di+1 placed on the equivalence classes given
with Pi+1 are monochromatic. In particular, the color of an embedding of
A into D0 depends only on the equivalence classes given by respective Pj ,
so the coloring p induces coloring:
p : (C
A=E ) ! f1; ::; rg:
The choice of the structure C provides B0
C, B0 u B=E such that
0
D0
p (B
-equivalence classes given by B0 contain
A=E ) = const. Clearly, E
B 2O EK such that B
D0
D, B u B and p (B
A ) = const. This
veri…es the Ramsey property for the class CE[K ].
((=)We suppose that K does not satisfy RP in order to show that the
class CE[K ] does not satisfy RP. Since K is not a Ramsey class there is a
natural number r, and there are structures A, B 2 K such that (B
A ) 6= ;
and
C
( ) For all C 2K satisfying (C
B ) 6= ;, there is a coloring p : (A ) !
0
C
C
f1; ::; rg such that for every B 2 (B ) we have p (A ) 6= const.
The structures A and B can be treated as members of the class CE[K ]
in which each equivalence class has exactly one element. Let D be a given
RELATIONAL QUOTIENTS
21
D=E
structure such that D = (D; fRiD gi2I ; E D ; D ) 2 CE[K ] and ( B ) 6= ;.
D=E
We consider the coloring p : ( A ) ! f1; ::; rg that veri…es statement ( )
for K and we de…ne the coloring
p : (D
A ) ! f1; ::; rg;
p(A0 ) = p(A0 =E):
where we consider A0 =E as a substructure of D=E given by the E D equivalence classes on which A0 is placed. Since the p-color of A0 depends
only on the E D -equivalence classes incident with A0 , it is straightforward
B0
to see that for all B0 2 (D
B ) we have p (A ) 6= const.
The previous theorem is a generalization of a result from [16] in the
sense that partial orderings are replaced with the more general form of
relational structures. In order to obtain the corresponding result in the
case of structures obtained by putting a structure on each equivalence class
we need the following.
Theorem 12. [26]Let l and r be natural numbers, and let (Ki )li=1 be a
sequence of Ramsey classes of …nite structures. Let (Ai )li=1 and (Bi )li=1 be
two sequences of structures such that Ai 2 Ki and Bi 2 Ki for all i
l.
l
Then there is a sequence (Ci )i=1 of structures such that Ci 2 Ki for all i l
and for any coloring
p : f(A01 ; A02 ; :::; A0l ) : (8i
i
l)[A0i 2 (C
A )]g ! f1; :::; rg;
i
there is a sequence of structures (B0i )li=1 satisfying: B0i 2 (C
Bi )
i l and
p f(A01 ; A02 ; :::; A0l ) : (8i
B0
l)[A0i 2 (Ai )](8i
Ci for all
l)g = const:
Using the Erdös-Rado arrow notation, for a sequence (Ci )li=1 of the structures that satis…es the conclusion of the previous theorem we write:
1 ;A2 ;:::;Al )
:
(C1 ; C2 ; :::; Cl ) ! (B1 ; B2 ; :::; Bl )(A
r
Note that Theorem 10 is the special case of Theorem 12 when we take
the sequence (Ki )li=1 such that the structures from these classes have no
relations on themselves.
22
MIODRAG SOKIĆ
Theorem 13. Let K be a class of …nite relational structures in a signature
LI = fRi gi2I . Let L be a class of …nite relational structures in a signature
LJ = fRj gj2J . Let K be an ordered expansion of K. Let L be an ordered
expansion of L such that L has JEP. If C[L ]E[K ] is the class of structures
given by De…nition 6, then K and L satisfy RP i¤ C[L ]E[K ] satis…es RP.
Proof. ()) We suppose that K and L satisfy RP in order to verify RP
for the class C[L ]E[K ]. Let A = (A; fRiA gi2I ; E A ; fRjA gi2J ; A ) and B =
(B; fRiB gi2I ; E B ; fRjB gi2J ; B ) be two structures from the class C[L ]E[K ]
such that (B
A ) 6= ;, and let r be a …xed natural number. The structures
A=E
B=E
A=E = (A=E; fRi gi2I ; A=E ) and B=E = (B=E; fRi gi2I ; B=E ) belong to the Ramsey class K , so there is a structure C = (C; fRiC gi2I ; C
) 2 K such that:
:
C ! (B=E)A=E
r
We de…ne the structure D = (D; fRiD gi2I ; E D ; fRjD gi2J ; D ) 2 C[L ]E[K ]
D=E
such that D=E = (D=E; fRi gi2I ; D=E ) = C. Each point from the set
C will be replaced with one E D -equivalence class which carries a structure
from L . The relations fRiD gi2I are de…ned such that for all i 2 I, j ni ,
[xj ]E D = cj :
RiD (x1 ; :::; xni ) , RiC (c1 ; :::; cni ):
The linear ordering D is de…ned so that each E D -equivalence class is an
interval with respect to D , and for all [y1 ]E D = c1 , [y2 ]E D = c2 , c1 6= c2 :
y1
D
y2 , c1
C
c2:
In order to de…ne each E D -equivalence class and relations fRjD gi2J we de…ne
a sequence fDs gas=0 of structures from C[L ]E[K ]. Finally, we take D = Da .
We list all embeddings A=E ,! C as fAl gal=1 , and we list all embeddings
B=E ,! C as fBl gbl=1 . Therefore a and b denote the size of the sets (C
A=E )
C
and (B=E ) respectively.
Now we de…ne D0 = (D0 ; fRiD0 gi2I ; E D0 ; fRjD0 gi2J ; D0 ) 2 C[L ]E[K ]
D =E
such that D0 =E = (D0 =E; fRi 0 gi2I ; D0 =E ) = C. We consider structures
B
B
B0;l = (B0;l ; fRi 0;l gi2I ; E B0;l ; fRj 0;l gi2J ; B0;l ) from C[L ]E[K ] such that
RELATIONAL QUOTIENTS
B
23
=E
B0;l =E = (B0;l =E; fRi 0;l gi2I ; B0;l =E ) = Bl . JEP for the class L implies
that we may take D0 such that B0;l D0 for all 1 l b.
Suppose that we have Dt and we want to de…ne Dt+1 for 0
t < a.
Then we denote by Dt At+1 the restriction of the structure Dt to E Dt equivalence classes given by At+1 , and we denote by Dt Act+1 the restricion
of Dt to the other E Dt -equivalence classes. Denote by fDt;k gpk=1 the structures from L de…ned on E Dt -equivalence classes. Note that in this list we
may have structures that ocur more then once. Without loss of generality
we may assume that the linear ordering Dt induces the linear ordering
Dt
Dt;p . We consider the structure
on fDt;k gpk=1 such that Dt;1 Dt
k p
A in a similar way. Let fA gk=1 be the list of the structures from L de…ned by E A -equivalence classes. Again, this is a list of structures that can
appear more then once. Without loss of generality, we may assume that
the linear ordering A induces the linear ordering on fAk gpk=1 such that
!
A
A1 A
Ap . By Theorem 12 there is a sequence E =fEt;k gpk=1 of
structures from L such that:
(Et;1 ; :::; Et;p ) ! (Dt;1 ; :::; Dt;p )(A
r
1 ;:::;Ap )
:
!
The sequence E de…nes structure Et from C[L ]E[K ] such that Et =E = At+1 .
The structures Et and Dt Act+1 de…ne Dt+1 .
We claim that D = Da is such that
Da ! (B)A
r :
To see this, consider a coloring:
p : (D
A ) ! f1; ::; rg:
Note that each embedding of the structure A into the structure D must
be placed on the collection of E D -equivalence classes given by one of the
(Al )al=1 . The choice of the sequence fDs gas=0 tell us that there is some Fa 1
such that
F
F
(1) Fa 1 = (Fa 1 ; fRi a 1 gi2I ; E Fa 1 ; fRj a 1 gj2J ; Fa 1 ) 2 C[L ]E[K ],
Fa 1 D a ,
(2) Fa 1 =E = C & Fa 1 = Da 1 ,
(3) all embedding of A into Fa 1 placed on the equivalence classes given
by Aa are monochromatic.
24
MIODRAG SOKIĆ
In general, there is a sequence of structures (Fs )as=01 from C[L ]E[K ] such
that Fi
Fi+1 for all 0
i < a, Fa 1 =E = C & Fa 1 = Da 1 , and
all embeddings of A into Fi+1 placed on the equivalence classes given with
Ai+1 are monochromatic. In particular, the color of an embedding of A
into F0 depends only on the equivalence classes given by respective Al , so
the coloring p induces the coloring:
p : (C
A=E ) ! f1; ::; rg:
The choice of the structure C provides B0
C, B0 u B=E such that
D0
B0
p (A=E ) = const. Clearly, E -equivalence classes given by B0 contain
B 2C[L ]E[K ] such that B D0 D, B u B and p (B
A ) = const.
((=)We suppose that K does not satisfy RP in order to show that the
class C[L ]E[K ] does not satisfy RP. Since K is not a Ramsey class there is
a natural number r, and there are structures A, B 2 K such that (B
A ) 6= ;
and
C
( ) For all C 2K satisfying (C
B ) 6= ;, there is a coloring p : (A ) !
C
f1; ::; rg such that for every B0 2 (C
B ) we have p (A ) 6= const.
The structures A and B can be treated as members of the class C[L ]E[K ]
in which each equivalence class has exactly one element and each one point
equivalence class carries the same structure from L . Let D be a given
structure such that D = (D; fRiD gi2I ; E D ; fRjD gj2J ; D ) 2 C[L ]E[K ] and
D=E
D=E
( B ) 6= ;. We consider the coloring p : ( A ) ! f1; ::; rg that veri…es
statement ( ) for K , and we de…ne the coloring
p : (D
A ) ! f1; ::; rg;
p(A0 ) = p(A0 =E):
where we consider A0 =E as a substructure of D=E given by the E D equivalence classes on which A0 is placed. Since the p-color of A0 depends
only on the E D -equivalence classes incident with A0 , it is straightforward
B0
to see that for all B0 2 (D
B ) we have p (A ) 6= const.
Since L can be seen as subclass of C[L ]E[K ] when we consider only
structures with one equivalence class, it follows that RP for the class C[L ]E[K ]
implies RP for the class L .
Note that Theorem 11 is a special case of Theorem 13 where we consider
the class of structures without any relations.
RELATIONAL QUOTIENTS
25
In contrast with the second part of the proof of Theorem 11, we are not
able to use a similar argument for the class OEK.
Lemma 9. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let
be a binary relational symbol such that 2
= L, and let
OEK be a class of …nite relational structures as given by De…nition 2. If
there is A 2 K such that Aut(A) is not a one-element group, then OEK
does not satisfy RP.
Proof. Let A be the underlying set of the structure A, and let i : A ! A,
1 i r , be a list of all automorphism of the structure A. Note that r > 1
because Aut(A) is not trivial. Let A be a linear ordering on the set A.
Then there are linear orderings i A on the set A such that for all x, y 2 A
we have x i A y () i 1 (x) A i 1 (y) for all 1 i r. For i 6= j we have
iA
6= j A . Note that we may view the structure A as a member of the
class EK and the structures (A; A ) and (A; i A ),1 i r, as members
of the class OEK. We consider a structure B 2 EK with underlying set B
such that B=E = A and for all b 2 B we have j[b]E B j = r. Therefore, there
are sets B1 , B2 ,...,Br such that:
(1) B = B1 [ ::: [ Br ; jAj = jB1 j =
= jBr j;
(2) i 6= j ) Bi \ Bj = ;;
(3) (8i 2 f1; ::; rg)(8b 2 Bi )(j[b]E B \ Bi j = 1);
On the set B, we put a linear ordering B such that:
(1) (8i; j 2 f1; ::; rg)(8bi 2 Bi )(8bj 2 Bj )(i < j ) bi B bj );
A
(2) (8i 2 f1; ::; rg)(8bi ; b0i 2 Bi )(bi B b0i () [bi ]E B
[b0i ]E B );
Consequently, (B; B ) 2 OEK and there are at least r substructures of
(B; B ) isomorphic with (A; A ). Note that the structure B is obtained
by placing r-many copies of A of di¤erent order type. The substructure of
(B; B ) given on the set Bi is isomorphic with (A; A ) by the map which
(B; B )
sends bi 2 Bi into 1 ([bi ]E B ) 2 A. In particular, (A; A ) 6= ;.
We claim that there is no (C; C ) 2 OEK such that C 2 EK and
(A; A )
(C; C ) ! (B; B )r
. Let be an arbitrary linear ordering of the un0
derlying set of the structure C=E. Let (A0 ; A ) 2 OEK be a structure with
0
0
underlying set A0 and linear ordering A such that (A0 ; A ) = (A; A )
0
and (A0 ; A ) (C; C ). Then there is a substructure (A"; A" ) (C; C )
0
such that for every a" 2 A" we have j[a"]E A" j = j[a"]E C j and (A0 ; A )
26
MIODRAG SOKIĆ
(A"; A" ). Moreover, the linear ordering
induces the linear ordering
0
of the set f[a"]E A" : a" 2 A"g. The linear ordering A induces the linear
ordering 0 on the set f[a"]E A" : a" 2 A"g such that for a, b 2 A0 we have
0
[a]E A"
[b]E A" , a
A0
b:
There is an automorphism of the structure A"=E which sends
Since A"=E = A , we may view
as A and 0 as one of
therefore as one of i ’s:We de…ne a coloring
p:
p((A0 ;
(C;
(A;
A0
C
A
)
)
! f1; :::; rg;
)) = i ()
=
i:
Now, it is straightforward to see that there is no (B;
that p
(B;
(A;
B
A
)
)
into 0 .
iA
, and
B
)2
(C;
(B;
C
B
)
)
such
= const, so OEK does not satisfy RP.
Remark 1. Note that in the previous Lemma we do not assume that the
class K satis…es HP, JEP or AP.
Let K be a class of …nite relational structures in a signature LI = fRi gi2I
with arity fni gi2I . We say that K satis…es the one-point property if whenever (A; fRiA gi2I ) 2 K and a 2 A then (fag; fRiA \ fagni gi2I ) 2 K. We say
that K satis…es the two-point property if there are A = (A; fRiA gi2I ) 2 K,
jAj = 1, and B 2 K such that j B
A j > 1.
Lemma 10. Let K be a class of …nite relational structures in a signature
LI = fRi gi2I . Let L be a class of …nite relational structures in a signature LJ = fRj gj2J . Assume that L satis…es the one-point and two-point
property. Let OLEK be the class of structures given by De…nition 5. If
K contains a non rigid structure or for every natural number k there are
structures A and B in L such that B contains at least k mutally disjoint
copies of A then OLEK does not satisfy RP.
Proof. Let A 2 K be such that jAut(A)j = a > 1. If there is B 2 L such
that the underlying set of B has at least a elements then we may proceed
as in the proof of Lemma 9. Otherwise for every B = (B; fRjB gj2J ) 2 L
we have jBj < a. In particular we may take B and A to verify the twopoint property for L, i.e. that j B
A j > 1. Note that the RP for OLEK
implies RP for OL; see Lemma 3 for de…nition of the class OL. We will
RELATIONAL QUOTIENTS
27
show that OL does not satisfy the RP. Suppose there is C 2 OL such
that C ! (B )A
a where B and A are some ordered expansions of B and
B
A respectively. Note that j B
j since A is a structure with
A j = j A
a one-point underlying set. We consider : C
! f1; :::; ag such that
A
(A) = i i¤ A is de…ned on the point a which is i-th element in C. Since
each structure in L has at most a points and B contains at least two copies
B
of A , there is no B 2 C
such that
= const. Therefore OL
B
A
does not satisfy RP, and OLEK does not satisfy the RP.
Remark 2. Note that in the previous Lemma we do not know if L satis…es
any of the following properties: HP,JEP and AP.
In order to examine RP for the class OEK when the class K from the
statement of Lemma 9 has rigid structures, we need to recall the de…nition
of -colored sets from [1]. Let
= f1; ::; ng where n is a given natural
number. Then the structure A = (A; <; f ), where A is a nonempty set with
linear ordering < and a function f : A ! , is called an -colored set. The
linear ordering < is called the underlying linear ordering of the -colored
set A. An embedding of -colored sets (A; <A ; fA ) and (B; <B ; fB ) is a
map F : A ! B such that for all x, y 2 A we have:
F (x) = f (F (x));
A
x < y , F (x) <B F (y):
An embedding which is a bijection is called an isomorphism; and we use
the symbol =.
Theorem 14. [1]Let n be a given natural number. Then the class of …nite
-colored sets satis…es RP.
Theorem 15. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let
and E be binary relational symbols such that 2
= L,
E 2
= L, and let OEK be a class of …nite relational structures as given by
De…nition 2. We assume that all A 2 K are rigid, i.e. Aut(A) is a oneelement group. If the class K satis…es RP, then OEK satis…es RP.
Proof. Let A = (A; fRiA gi2I ; E A ; A ) and B = (B; fRiB gi2I ; E B ; B ) be
from OEK such that (B
A ) 6= ;, and let r be a given natural number. Then
28
MIODRAG SOKIĆ
A=E
B=E
A=E = (A=E; fRi gi2I ) and B=E = (B=E; fRi gi2I ) are structures
from the Ramsey class K, so there is C = (C; fRiC gi2I ) 2 K such that
C ! (B=E)A=E
:
r
We use the structure C to obtain D = (D; fRiD gi2I ; E D ; D ) 2 OEK. Each
E D -equivalence class is obtained by replacing each point from C with a set.
Placing linear ordering D on D is the hardest part of the proof. The rest
is motivated by partite construction of Nešetµril-Rödl, see [17] and [18].
We list all substructures inside C isomorphic with A=E as (Aj )aj=1 and all
substructures inside C isomorphic with B=E as (Bj )bj=1 . Recursively, we deC
…ne a sequence of structures (Cj )aj=0 of the form Cj = (Cj ; fRi j gi2I ; E Cj ; Cj
) 2 OEK such that for all j, j 0 holds:
C =E
(Cj =E; fRi j
C 0 =E
gi2I ) = (Cj 0 =E; fRi j
gi2I ):
C0 is a disjoint union: C0 = [bl=1 C0;l . Set C0;l is placed on the E D equivalence classes given by Bl such that:
(1) E C0;l -equivalence classes are given by E D ,i.e. xE C0;l y , xE D y;
C
(2) The relations fRi 0;l gi2I are inherited from relations fRiC gi2I , i.e. for
i
2 C0;i , [xj ]E C0;l = cj holds:
i 2 I, (xj )nj=1
C
Ri 0;l (x1 ; :::; xni ) , RiC (c1 ; :::; cni );
(3) The linear ordering
C0;l
on C0;l is such that:
C
Co;l = (Co;l ; fRi 0;l gi2I ; E C0;l ;
C0;l
) u B:
fRiC0 gi2I
There is only one way to de…ne relations
and E C0 on the set C0
C0
such that (C0 =E; fRi gi2I ) = C, and it comes from relations fRiC gi2I and
from the fact that each point from C produces one E D -equivalence class.
The linear ordering C0 is de…ned such that:
C0
C0;1
jC0;l =
C0
C0;2
C0;l
C0
for each l,
:::
C0
C0;b :
In this way we have a structure from OEK that contains copies of B placed
on b disjoint sets.
RELATIONAL QUOTIENTS
29
Assume that we have the structure Ck . Then, we de…ne Ck+1 as follows.
The restriction of the structure Ck to E Ck -equivalence classes given by Ak+1
we denote with:
C jA
Ck jAk+1 = (Ck jAk+1 ; fRi k k+1 gi2I ; E Ck jAk+1 ; Ck jAk+1 ):
We denote E Ck jAk+1 -equivalence classes with (es )gs=1 and E A -equivalence
classes with (e0s )gs=1 . Without loss of generality, we assume that all embeddings of A into Ck jAk+1 must send class e0s into class es . This follows from
A=E
the fact that (A=E; fRi gi2I ) is a rigid structure, and K is the class of rigid
structures. In the following we consider -colored sets with = f1; :::; gg.
We code Ck jAk+1 using the -colored set (Ck jAk+1 ; Ck jAk+1 ; fCk jAk+1 ), such
that for x 2 Ck jAk+1 we have
fCk jAk+1 (x) = s , x 2 es :
Similarly we code A with the -colored set (A; A ; fA ) such that for y 2 A
we have
fA (y) = s , y 2 e0s :
By Theorem 14, there is an -colored set (Fk ;
(Fk ;
Fk
; fFk ) ! (Ck jAk+1 ;
Fk
Ck jAk+1
Fk
; fFk ) such that
; fCk jAk+1 )(A;
r
(Fk ; fRiFk gi2I ; E Fk ;
A ;f
A)
:
Fk
) 2 OEK in
; fFk ) gives Fk =
The structure (Fk ;
the following way. If fFk (x) = s for x 2 Fk , then x is placed on the E D equivalence class given by es . The relations fRiFk gi2I are de…ned such that
for x1 ,...,xni 2 Fk we have
C jAk+1
RiFk (x1 ; :::; xni ) () Ri k
(y1 ; :::; yni );
where y1 ,...,yni 2 Ck jAk+1 and
fFk (y1 ) = fCk jAk+1 (y1 ); :::; fFk (yni ) = fCk jAk+1 (yni ):
k
List all embeddings Ck jAk+1 ,! Fk as (Clk )nl=1
. Over each Clk we can amalgamate the structure Clk isomorphic with Ck by preserving E D -equivalence
classes such that it intersects Fk exactly over Clk . For a detailed explanation
of this amalgamation see [25]. Each structure Clk has the underlying set Ckl .
0
For l 6= l0 the structures Clk and Clk can intersect only over points inside
the set Fk . The set Ck+1 is the union:
30
MIODRAG SOKIĆ
C
k
Ck+1 = Fk [ ([nl=1
Ckl )
and the relations fRi k+1 gi2I and E Ck+1 are de…ned in obvious way. The
l
k
linear orderings f Fk g [ f Ck gnl=1
generate a partial ordering k on the set
Ck+1 and for Ck+1 we can take an arbitrary linear extension of k , so the
structure Ck+1 is de…ned.
In the end we have Ca , and we claim that:
Ca ! (B)A
r :
a
Now, we take D = Ca . So let p : (C
A ) ! f1; ::; rg be a given coloring. The
^ a 1 2 OEK such
construction of the structure Ca implies that there is C
that:
^a
C
^ a 1 Ca
u Ca 1 ; C
^ a 1 placed on the E D -equivalence
and the coloring of embeddings A ,! C
classes given by Aa is constant. Recursively, we de…ne a sequence of struca 1
^ j )j=0
tures (C
such that
1
^ j 2 OEK; C
^ j u Cj ; C
^j C
^ j+1
C
^ j placed on the E D -equivalence
and the coloring of embeddings A ,! C
classes given by Aj+1 is constant. In the end
^ 0 2 OEK; C
^ 0 u C0 ; C
^ 0 Ca
C
^ 0 depends only on the E D -equivalence
and the coloring of an embedding A ,! C
classes on which embedding is placed. Therefore, we obtain an induced coloring: p^ : (C
A ) ! f1; ::; rg. The choice of the structure C implies existence
of a structure B0 u B=E such that
0
p^ (B
A0 ) = const:
The equivalence classes given by B0 provide a structure B 2 OEK, B u B,
^ 0 Ca and p (B
B
C
A ) = const, so RP is veri…ed for the class OEK.
In order to examine the Ramsey property for the class OLEK, we need
to introduce new concepts. Let L be a class of …nite relational structures in
signature LJ = fRj gj2J with arity fnj gj2J . Let n be a natural number and
RELATIONAL QUOTIENTS
31
let fIi gni=1 be a list of unary relational symbols which do not belong to LJ .
Then we denote by Ln the class of structures, in signature LJ [ fIi gni=1 , of
the form (A; fIiA gni=1 ; fRjA gj2J ) such that for all 1 i; i0 n we have:
S
(1) A = ni=1 fa 2 A : IiA (a)g,
(2) i 6= i0 ) fa 2 A : IiA (a)g \ fa 2 A : IiA0 (a)g = ;
n
(3) If Ai = fa 2 A : IiA (a)g then (Ai ; fRjA \ Ai j gj2J ) 2 L,
(4) For all a1 ; :::; anj 2 A with RjA (a1 ; :::; anj ) there is i such that IiA (a1 ),...,
IiA (anj ).
If we add arbitrary linear orderings to structures from the class Ln then we
obtain the class OLn . It contains structures in the signature LJ [fIi gni=1 [f
g where
is a binary symbol not in LJ . Structures from OLn are of the
form (A; fIiA gni=1 ; fRjA gj2J ; A ) such that (A; fIiA gni=1 ; fRjA gj2J ) is in Ln
and A is a linear ordering on the set A. Then we say that the class L
is n-adaptable if OLn satis…es the RP. If L is n-adaptable for all natural
numbers n then we say that L is adaptable. We introduce adaptable classes
in order to transfer the proof of Theorem 15 into a corresponding proof for
the class OLEK. More precisely, we make the recursive step in the proof of
Theorem 15 possible for the class OLEK under assumption of adaptability.
Using the same idea of the proof of Theorem 15 we obtain the following.
Theorem 16. Let K be a class of …nite rigid relational structures in a
signature LI = fRi gi2I . Let L be a class of …nite relational structures in
a signature LJ = fRj gj2J . Let OLEK be the class of structures given by
De…nition 5. If the class K satis…es RP and L is adaptable then OLEK
satis…es RP.
Note that Theorem 15 can be obtained as a corollary of Theorem 16 if
we take L to be the class of …nite sets in empty signature. In the following
we show the importance of the adaptability in Theorem 16.
Lemma 11. Let K be a class of …nite rigid relational structures in a signature LI = fRi gi2I . Let L be a class of …nite relational structures in a
signature LJ = fRj gj2J . Let OLEK be the class of structures given by De…nition 5. If L is not adaptable and K contains structures of arbitrarily
large …nite cardinality then OLEK does not satis…es RP.
32
MIODRAG SOKIĆ
Proof. Let n be the smallest natural number such that L is not n-adaptable.
Then there are structures
A = (A; fIiA gni=1 ; fRjA gj2J ;
A
) and B = (B; fIiB gni=1 ; fRjB gj2J ;
B
)
from OLn and there is a natural number r such that there is no B 2OLn
with the property C ! (B)A
r . Since n is the smallest such number, we have
for each 1 i n that there are a 2 A and b 2 B such that IiA (a) and IiB (b).
For each 1 i n we denote by Ai and Bi structures from L induced on
the sets fa 2 A : IiA (a)g and fb 2 B : IiB (b)g respectively. By assumption
on the class K there is D 2K with underlying set D = fd1 ; :::; dm g such that
m > n. We consider structures
DA =(DA ; fRiDA gi2I ; E DA ; fRjDA gi2J ;
DB =(DB ; fRiDB gi2I ; E DB ; fRjDB gi2J ;
DA
) and
DB
)
from OLEK. The equivalence classes in DA and DB given by di contain
structures Ai and Bi respectively for 1
i
n. For i > n, equivalence
classes given by di contain some …xed structure H from L in both cases DA
and DB . Let DA jn and DB jn denote subsets of DA and DB , respectively,
given by equivalence relations obtained from fd1 ; :::; dn g. Then there are
bijections A : A ! DA jn and B : B ! DB jn which map fa 2 A : IiA (a)g
i
and fb 2 B : IiB (b)g to equivalence classes given by di for all 1
n. Moreover we assume that A and B give isomorphisms between Ai
and Bi and corresponding structures de…ned on equivalence classes given
by di for all 1
i
n. Then we take DA and DB such that their
restrictions to DA jn and DB jn are given by A ( A ) and B ( B ). We take
: DA n(DA jn) ! DB n(DB jn) to be a bijection which sends the equivalence
class given by di into the equivalence class given by di , i > n. Also we assume
that is a bijection of the structures isomorphic to H which are placed on
equivalence classes given by the same di . Then we take that the restriction
of the linear orderings DA and DB to DA n(DA jn) and DB n(DB jn) are
isomorphic over . Let a and a0 be elements from DA which belong to
equivalence classes given by di and di0 . If i0 > n and i
n then we take
DA
0
di
di0 . Similarly if b and b are elements from DB which belong to
equivalence classes given by di and di0 where i0 > n and i
n then we
DB
D
D
A
B
take di
di0 . Therefore the linear orderings
and
are de…ned
B
and we have D
=
6
;.
We
claim
that
there
is
no
D
in
OLEK
such that
C
DA
RELATIONAL QUOTIENTS
33
A
DC ! (DB )D
r . Otherwise there is DC which satis…es the previous arrow
notation. Note that every copy of DA and DB inside DC is placed on the
equivalence classes given by some copy of D inside F 2K where F = DC =E.
F
We list D
= fD1 ; :::; Dk g. We de…ne
:
DC
DA
! f1; :::; rg;
by de…ning i to be the restriction of given by the structure Di . This
is well de…ned since the structures in K are rigid. It is enough to show
that none of the i ’s contains a monochromatic copy of DB . According to
our choice of DA and DB , we may treat the coloring i as a coloring of
the structures in OLm . Then we may choose i such that two copies of
DA have the same color if their restriction to the set given by the …rst n
indicators are the same. That means that we may view i as a coloring of
the structures A in OLn , and we are looking for a monochromatic B. Since
OLn is not a Ramsey class we may choose i to be without a monochromatic
structure.
Lemma 12. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let
and E be binary relational symbols such that 2
= L,
E 2
= L, and let OEK be a class of …nite relational structures as given by
De…nition 2. We assume that all A 2 K are rigid, i.e. Aut(A) is a oneelement group. If the class K does not satisfy RP then OEK does not satisfy
RP.
Proof. Let r be a natural number, and let A = (A; fRiA gi2I ) and B =
(B; fRiB gi2I ) be structures from K such that (B
A ) 6= ;. We assume that
C
for every C 2 K with (B ) 6= ; there is a coloring p : (C
A ) ! f1; :::; rg
B0
),
we
have
p
(
)
=
6
const.
Structures A
such that for every B0 2 (C
B
A
and B we may view as members of the class EK, in which every E A and
every E B equivalence class have exactly one element. Let (Aj )aj=1 be a list
of all structures Aj = (Aj ; fRiA gi2I ) 2 K such that Aj
B and Aj =
A
A. By adding an arbitrary linear ordering
to the set A, we get a
structure A = (A; fRiA gi2I ; E A ; A ) 2 OEK. We consider a structure B =
(B; fRiB gi2I ; E B ; B ) 2 OEK such that:
(1) B = [aj=1 Bj ;
(2) j 6= j 0 =) Bj \ Bj 0 = ;, for all j; j 0 2 f1; ::; ag;
34
MIODRAG SOKIĆ
(3) Each Bj is placed on the E B equivalence classes given by Aj such
that for all b 6= b0 from Bj we do not have bE B b0 ;
(4) If BjBj is a substructure of B given by the set Bj then we have
BjBj = A.
Now, let C = (C; fRiC gi2I ; E C ; C ) 2 OEK be such that (C
) 6= ;. Then,
B
C=E
C=E
C=E = (C=E; fRi gi2I ) 2 K, and there is coloring p : (A ) ! f1; :::; rg
0
C=E
such that for every B0 2 (B ), we have p (B
A ) 6= const. Using p, we de…ne
the coloring
p : (C
A ) ! f1; :::; rg;
p(A0 ) = p(Aj );
where Aj gives the E C classes on which structures A0 is placed. Now, it is
0
straightforward to check that for all B0 2 (C
) , we have p (B
) 6= const, so
B
A
OEK does not satisfy RP.
We are not able to drop the assumption on the Ramsey property for the
class K in Theorem 16.
Lemma 13. Let K be a class of …nite rigid relational structures in a signature LI = fRi gi2I . Let L be a class of …nite relational structures in a
signature LJ = fRj gj2J . Let OLEK be the class of structures given by De…nition 5. Assume that K does not satisfy RP and assume that for every
natural number k there are structures A and B in L such that B contains
at least k mutally disjoint copies of A. Then OLEK does not satisfy RP.
Proof. This is done in the similar way as the proof of Lemma 12. The
di¤erence is that we de…ne the sets Bj , see proof of Lemma 12, not to be
transversal but to be of the form C A.
5. Ordering property
Proposition 1. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let
be a binary relational symbol such that 2
= L, and let
K be a class of …nite relational structures in a signature L [ f g such that
K is an ordered expansion of K and K jL = K. We assume that CE[K ]
is the class of structures given by De…nition 3. Then K satis…es OP with
respect to K i¤ CE[K ] satis…es OP with respect to EK.
RELATIONAL QUOTIENTS
35
Proof. (=)) We assume that K satis…es OP with respect to K. For a
A=E
given A = (A; fRiA gi2I ; E A ) 2 EK, we have A=E = (A=E; fRi gi2I ) 2 K.
OP of the class K with respect to K implies existence of a structure B =
(B; fRiB gi2I ) 2 K such that for all linear orderings A=E and B on A=E
A=E
and B respectively with (A=E; fRi gi2I ; A=E ), (B; fRiB gi2I ; B ) 2 K
we have:
A=E
(A=E; fRi gi2I ; A=E ) ,! (B; fRiB gi2I ; B ):
We consider C = (C; fRiC gi2I ; E C ) 2 EK such that C=E = B and for all
c 2 C:
j[c]E C j = maxfj[a]E A j : a 2 Ag:
Let
be a given linear ordering on C, and let A be some linear ordering
on A. By OP for K , there is an embedding:
C
A=E
e : (A=E; fRi
gi2I ;
A=E
) ! (B; fRiB gi2I ;
B
C=E
) = (C=E; fRi
gi2I ;
C
);
and according to the size of the E C -equivalence classes in C, there is an
embedding f : A ! C such that fE = e, so OP is veri…ed for CE[K ] with
respect to EK.
((=) We assume that CE[K ] satis…es OP with respect to EK, but
K does not satisfy OP with respect to EK. Note that the embedding
f : A ! B, where A and B are from CE[K ], induces an embedding
fE : A=E ! B=E of structures from EK. Therefore, if CE[K ] satis…es
OP with respect to EK, then K would satisfy OP with respect to K. This
contradicts our assumption, so OP is veri…ed in this case.
Proposition 2. Let K be a class of …nite relational structures in a signature
LI = fRi gi2I . Let L be a class of …nite relational structures in a signature
LJ = fRj gj2J which satis…es JEP. Let K be an ordered expansion of K.
Let L be an ordered expansion of L. Let C[L ]E[K ] be the class of structures given by De…nition 6, and let LEK be the class of structures given by
De…nition 4. Then the classes K and L satisfy OP with respect to K and
L respectively i¤ C[L ]E[K ] satis…es OP with respect to LEK.
Proof. (=))Let A be a given structure from LEK. We assume that A is
represented in the form C; (Dc )c2C where C is a structure from K with
36
MIODRAG SOKIĆ
underlying set C and Dc 2L for all c 2 C; see the paragraph after De…nition
4. Since L satis…es OP with respect to L then for each c 2 C there is
Dc 2L which veri…es OP for Dc . JEP for the class L implies that there is
D 2L such that Dc ,! D. Since K satis…es OP with respect to K there is
E 2L, with underlying set E, which veri…es OP for C. We claim that the
structure E; (Fe )e2E where Fe = D for all e 2 E veri…es OP for A. Let
A
be a linear orderingDsuch that (A; A ) 2 C[L E]E[K ]. Then the structure
(A; A ) is of the form (C; C ); ((Dc ; Dc ))c2C where (C; C ) 2 K and
(Dc ; Dc ) 2 LD for c 2 C; see the paragraph
after De…nition 6. We consider
E
Fe
E
the structure (E;
); ((Fe ;
))e2E from C[L ]E[K ]. By the choice of
the structure E there is an embedding from (C; C ) into (E; E ). By the
choice of the structure D there is an embedding of (Dc ; Dc ) into (Fe ; Fe )
for all
we have an embedding from (A; A )
E
D c 2 C and all e 2 E. Therefore
into (E; E ); ((Fe ; Fe ))e2E .
((=)Suppose that L does not satisfy OP with respect to L. Since L satis…es JEP, there is D 2 L and a linear ordering C such that (D; D ) 2 L
but for all E 2 L and all linear orderings E with (E; E ) 2 L there
is no embedding from (D; D ) into (E; E ). Then we consider an arbitrary C 2 K, with underlying set C, andDA = C; (Dc )c2C fromE LEK
where Dc = D for all c 2 C. We consider (C;
C
); ((Dc ;
Dc
))c2C
from
C[L ]E[K ] such that (Dc ; Dc ) = (D; D ) for all c 2 C. Let
D
E
Fe
E
(E;
); ((Fe ;
))e2E 2 C[L ]E[K ]
be such that a linear orderings Fe are such that there is no embedding
from (D; D ) into (Fe ; Fe ). Then there is no embedding from
D
E
D
E
(C; C ); ((Dc ; Dc ))c2C ,! (E; E ); ((Fe ; Fe ))e2E ;
so C[L ]E[K ] does not satisfy OP with respect to LEK.
D Suppose that K doesEnot satisfy
D OP with respect to K.
E Let A =
Dc
Fe
C
E
(C;
); ((Dc ;
))c2C and B = (E;
); ((Fe ;
))e2E be two structures from C[L ]E[K ]. Then any embedding from A into B induce an embedding from (C; C ) into (E; E ). So if (C; C ) is a witness that K does
RELATIONAL QUOTIENTS
37
not satisfy OP with respect to K, then A is a witness that C[L ]E[K ] does
not satisfy OP with respect to LEK.
Lemma 14. Let K be a class of …nite relational structures in a signature
L = fRi gi2I . Let
and E be binary relational symbols such that 2
= L,
E 2
= L, and let OEK be a class of …nite relational structures as given by
De…nition 2. Then OEK does not satisfy OP with respect to EK.
Proof. Note that for A 2 OEKn CE[K ] and B 2 CE[K ] there is no embedding A ,! B, so OEK does not satisfy OP with respect to EK.
Lemma 15. Let K be a class of …nite relational structures in a signature
LI = fRi gi2I . Let L be a class of …nite relational structures in a signature
LJ = fRj gj2J . Let K be an ordered expansion of K. Let L be an ordered
expansion of L. Let OLEK be the class of structures given by De…nition 5
and let LEK be the class of structures given by De…nition 4. Then the class
OLEK does not satisfy OP with respect to LEK.
Proof. This follows from the fact that for A 2 OLEKn C[L ]E[K ] and
B 2 C[L ]E[K ] there is no embedding A ,! B.
6. Dynamics
In this section we present a few topological implication of results from
Sections 4 and 5.
Let G be a topological group. A G-‡ow X is a continuous action G X !
X on a compact Hausdor¤ space X. A G-‡ow X is minimal if each of its
orbits is dense, i.e. for all x 2 X we have X = G x. If every minimal
G-‡ow X is a point then we say that G is an extremely amenable group.
Equivalently, G is extremely amenable if every G-‡ow X contains a …xed
point x, i.e. we have gx = x for all g 2 G. The following result shows that
among all minimal G-‡ows there is a largest one.
Theorem 17. [2]Given a topological group G, there is a minimal G-‡ow X
with the following property: For any minimal G-‡ow Y there is a surjective
continuos map : X ! Y such that g (x) = (g x) for all g 2 G,
x 2 X. Moreover the G-‡ow X is uniquely determined up to isomorphism,
i.e. for any other G-‡ow X satisfying the same property there is a continuos
bijection : X ! X that g (x) = (g x) for all g 2 G, x 2 X.
38
MIODRAG SOKIĆ
The G-‡ow given by the previous theorem is called the universal minimal
G-‡ow.
A way to recognize extremely amenable groups among groups of automorphisms of Fraïssé structures is given in [11]. In addition to this, the
authors of [11] provide the technique for the calculation of universal minimal ‡ows of certain automorphism groups of Fraïssé structures. In the
following, we present material that we need for topological interpretation of
our combinatorial results from the previous sections.
If L is a countable structure, then we consider the group Aut(L) of its
automorphisms as a topological group with pointwise topology, see [3].
Theorem 18. [11]Let L be a Fraïssé ordered class and let L = F lim(L).
Then, the topological group Aut(L) with the pointwise topology is extremely
amenable i¤ L satis…es the Ramsey property.
In the case that the topological group is not extremely amenable, we will
try to calculate its universal minimal ‡ow. Let L0 and L1 be countable
signatures such that L0 L1 = L0 [ f g, 2
= L0 , where is a binary relational symbol. We assume that a Fraïssé class L1 , which is in the signature
L1 , is an order expansion of a Fraïssé class L0 , which is in the signature L0 .
Also, we assume that is interpreted as a linear ordering in each structure
from L1 . We consider Fraïssé limits: L0 = F lim(L0 ) and L1 = F lim(L1 ).
In terms of reduct we have L0 = L1 jV0 . The interpretation of the symbol
in L1 is as a linear ordering L1 . Without loss of generality we may assume
that structures L0 and L1 are de…ned on the set of natural numbers N.
The set of all linear orderings of natural numbers LO 2N N is a compact
Hausdor¤ space, see [3]. In particular L1 2 LO. Let G0 = Aut(L0 ) and
consider the logic action of G0 on LO, see [3]. The topological closure of
the orbit of S under logic action:
XS = G0 S LO;
is important for calculation of universal minimal ‡ows. Linear orderings
from XS we call S-admissible.
Theorem 19. [11]Let L1 and L0 be countable signatures such that L0 =
L1 nf g where
is a binary relational symbol. Let L1 be a reasonable
Fraïssé order class in L1 and let L0 be a Fraïssé class in L1 . We assume that
S0 = S1 jL0 and that Fraïssé limits S1 = F lim(S1 ) and S0 = F lim(S0 ) are
RELATIONAL QUOTIENTS
39
de…ned on the set of natural numbers. Let G0 = Aut(S0 ) be the topological
group with pointwise topology, and let XS1 be the set of L1 -admissible linear
orderings. Then we have:
(i) XL1 is the universal minimal G0 -‡ow i¤ L1 is a Ramsey class satis…es
the ordering property with respect to L0 .
(ii) XL1 is a minimal G0 -‡ow i¤ L1 satis…es the ordering property with
respect to L0 .
If K is a Fraïssé class of …nite relational structures in a signature L =
fRi gi2I , then I has to be countable.
Theorem 20. Let K be a Fraïssé class of …nite relational structures in
a signature L = fRi gi2I . Let
and E be binary relational symbols such
that 2
= L, E 2
= L. Let K be a Fraïssé class of …nite relational structures
in the signature L [ f g such that K is an ordered expansion of K and
K jL = K. We assume that EK, OEK and CE[K ] are classes of structures
given by De…nition 1, De…nition 2 and De…nition 3 respectively. Then:
(i) The Fraïssé limits EK = F lim(EK), OEK = F lim(OEK) and
CE[K ] = F lim(CE[K ]) are well-de…ned.
(ii) The topological group Aut(CE[K ]) is extremely amenable i¤ K satis…es RP.
(iii) The topological group Aut(OEK) is extremely amenable i¤ K is a
Ramsey class of …nite rigid structures.
(iv) The Aut(EK)-‡ow XOEK is not minimal.
(v) The Aut(EK)-‡ow XCE[K ] is minimal i¤ K satis…es OP with respect
to K.
(vi) XCE[K ] is the universal minimal Aut(EK)-‡ow i¤ K satis…es RP
and OP with respect to K.
Proof. (i) This follows from Lemmas 1, 3, 4 and 6.
(ii) This follows from Theorem 18 and Theorem 11.
(iii) This follows from Theorem 18, Lemma 9, Lemma 12 and Theorem
15.
(iv) and (v) This follows from Theorem 19 and Proposition 1.
(vi) This follows from Theorem 19, Theorem 11 and Proposition 1.
40
MIODRAG SOKIĆ
Theorem 21. Let K be a Fraïssé class of …nite relational structures in a
signature LI = fRi gi2I . Let L be a Fraïssé class of …nite relational structures in a signature LJ = fRj gj2J such that LI \ LJ = ;. Assume that L
satis…es SAP and that at least one of L or K satisfy SJEP. Let K be a
reasonable Fraïssé ordered expansion of K. Let L be a reasonable Fraïssé
ordered expansion of L. let LEK be the class of structures given by De…nition 4. Let OLEK be the class of structures given by De…nition 5. Let
C[L ]E[K ] be the class of structures given by De…nition 6 . Then:
(i) The Fraïssé limits LEK = F lim(LEK), OLEK = F lim(OLEK) and
C[L ]E[K ] = F lim(C[L ]E[K ]) are well-de…ned.
(ii) The topological group Aut(C[L ]E[K ]) is extremely amenable i¤ K
and L satisfy RP.
(iii) If K contains a non rigid structure then the topological group Aut(OLEK)
is not extremely amenable.
(iv) If K is a Ramsey class and L is adaptable then the topological group
Aut(OLEK) is extremely amenable.
(v) If for every natural number k there are structures A and B in L such
that B contains at least k mutally disjoint copies of A then Aut(OLEK) is
not extremely amenable.
(vi) The Aut(LEK)-‡ow XOLEK is not minimal.
(vii) The Aut(LEK)-‡ow XC[L ]E[K ] is minimal i¤ K and L satisfy OP
with respect to K and L respectively.
(viii) XC[L ]E[K ] is the universal minimal Aut(LEK)-‡ow i¤ K and L
satisfy RP and OP with respect to K and L respectively.
Proof. This is done in the same way as the proof of Theorem 20.
7. Finite topological spaces
We denote the class of …nite topological spaces by T and its elements
by (X; ) where is a topology on X. A topology on a …nite set X has
a base fUx : x 2 Xg where
\
Ux = fU 2 : x 2 U g;
and there is a binary relation
have
x
on X de…ned such that for x, y 2 X we
y , x 2 Uy :
RELATIONAL QUOTIENTS
41
Clearly, the relation is transitive and re‡exive, but it is not antisymmetric.
Consequently, is a quasi ordering on X and pair (X; ) is a qoset. In this
way we assign to each …nite topological space a qoset. It is also possible to
make an assignment in the opposite direction. Let (Y; ) be a qoset. Then
we have a collection of sets U = fUy : y 2 Y g such that
Uy = fz 2 Y : z
yg:
Clearly, U is a base of a topology on X. Note that this is 1 1 connection
between the class of …nite topological spaces and …nite qosets. Instead
of examining the class of …nite topological spaces with linear ordering we
may examine the class of …nite qosets with linear ordering. Also, there
is no countable signature describing …nite topological spaces, but we have
countable signature describing …nite qosets. We refer reader to [16] and [28]
for more details about …nite topological spaces.
Let Q be the class of …nite qosets and let P be the class of …nite posets.
According to De…nition 1 we have Q = EP.
For a given set A, the collection of all linear orderings on A is denoted
by lo(A). If is a given partial ordering on a set A, then the collection of
all linear extensions of the partial ordering is denoted by le( ). Then we
have the class of …nite posets with linear orderings:
P1 = f(A; ; ) : (A; ) 2 P; 2 lo(A)g;
and the class of …nite posets with linear extensions:
P2 = f(A; ; ) 2 P1 : 2 le( )g:
P is a Fraïssé, see [23], as well as P1 and P2 , see [25], so by Lemma 1 and
Lemma 4 Q = EP and Q0 = OEP are Fraïssé classes. If we take K = P1 or
K = P2 then by De…nition 3 we get classes Q1 = CE[P1 ] and Q2 = CE[P2 ]
which are also Fraïssé by Lemma 6. Note that we have three reasonable
Fraïssé ordered expansions of the class Q: Q0 , Q1 and Q2 . We also have
the corresponding Fraïssé limits: Q, Q0 , Q1 , Q2 and their automorphism
groups G, G0 , G1 , G2 respectively. The class P1 does not satisfy OP with
respect to P, but P2 satis…es OP with respect to P, see [25].
Theorem 22. (i) [24], [25] P1 is not a Ramsey class.
(ii) [20], [19], [5], [24], [25] P2 is a Ramsey class.
42
MIODRAG SOKIĆ
Applying the results from Section 4 and Section 5 to classes P, P1 and
P2 , we have the following consequence.
Corollary 1. (i) Q0 and Q1 are not Ramsey classes, but Q2 is a Ramsey
class( see [16]).
(ii) Q0 and Q1 do not satisfy OP with respect to Q, but Q2 satis…es OP
with respect to Q.
From Theorem 20 we get topological implications.
Corollary 2. (i) G0 and G1 are not extremely amenable groups, but G2
is an extremely amenable group.
(ii) XQ0 and XQ1 are not minimal G-‡ows, but XQ2 is the universal
minimal G-‡ow.
Remark 3. Without loss of generality, we may assume that the qoset Q =
(N; ) has the set of natural numbers as underlying set. On N, we may
de…ne the collection B = fUx : x 2 Ng where Ux = fy 2 N : y xg. Clearly,
B is a base of a topology T on N. The topological space (N; T ) contains any
…nite topological space as subspace, but it does not contain all countable
topological spaces as subspaces because its weight is countable. The qosets
Q0 , Q1 and Q2 may be viewed as topological space (N; T ) with an additional
linear ordering, so Corollary 1 and Corollary 2 can be stated in terms of
topological spaces. In this case G is actually the group of homeomorphisms
of the topological space (N; T ) with the pointwise topology, and G0 , G1 , G2
are its subgroups.
8. Finite metric spaces
We denote the class of …nite metric spaces with distances in the set of
rational numbers Q by M. Moreover M may be viewed as a class of …nite
structures in the signature L = fRi gi2Q , where each Ri is a binary relational
symbol. To a metric space (A; d) we assign a structure (A; fRiA gi2Q ) such
that for all x, y 2 A and all i 2 Q we have
RiA (x; y) , d(x; y) = i:
Adding linear orderings to …nite metric spaces we obtain the class of …nite
linearly ordered metric spaces:
M1 = f(A; fRiA gi2Q ; ) : (A; fRiA gi2Q ) 2 M; 2 lo(A)g:
RELATIONAL QUOTIENTS
43
By De…nition 1 the class R = EM corresponds to the class of …nite pseudometric spaces with rational distances. De…nition 2 give us a class R0 = OEM
of linearly ordered …nite pseudometric spaces. If we take K = M1 then
by De…nition 3 we get the class R1 = CE[M1 ] of convex linearly ordered
pseudometric spaces. It is known that M and M1 are Fraïssé classes, see
[11], such that M1 is a Ramsey class satisfying OP with respect to M,
see [15]. Lemma 1, Lemma 4 and Lemma 6 imply that R, R0 and R1 are
Fraïssé classes with limits R, R0 and R1 respectively. The automorphism
groups of R, R0 and R1 are H, H0 and H1 respectively, and the results of
Sections 4 and 5 give us the following.
Corollary 3. (i) R1 is a Ramsey class, but R0 is not a Ramsey class.
(ii) R1 satis…es OP with respect to R, but R0 does not satisfy OP with
respect to R.
We also have a topological implication of Theorem 20.
Corollary 4. (i) H1 is an extremely amenable group, but H0 is not an
extremely amenable group.
(ii) XR1 is the universal minimal H-‡ow, but XR0 is not a minimal H‡ow.
We derive a new class of …nite ordered metric spaces. Let S 6= ; be a
countable subset of (0; +1) such that inf S > 0 where inf S is the in…mum
of the set S. We denote the class of …nite metric spaces with distances in
the set S by MS . Structures from MS can be seen as relational structures
in the signature LS = fRs gs2S where each Rs is a binary symbol. A metric
space (A; d) from MS is coded by a relational structure (A; fRsA gs2S ) such
that for all x, y 2 A and all s 2 S we have
RsA (x; y) , d(x; y) = s:
If we enlarge the set S by adding a point q 2
= S to obtain S = fS; qg, then
we denote the class MS also by MfS;qg . Adding linear ordering to …nite
metric spaces from MS we obtain the class of …nite linearly ordered metric
spaces:
OMS = f(A; fRsA gs2S ; ) : (A; fRsA gs2S ) 2 MS ; 2 lo(A)g:
By De…nition 1 we get the class RS = EMS which corresponds to the class
of …nite pseudometric spaces with distances in the set S. Let q 2 R be
such that 0 < q < inf S. We may view the class RS as a subclass of
44
MIODRAG SOKIĆ
MfS;qg . When we want to view RS as a class of metric spaces then we will
denote it by M[S;q] : De…nition 2 gives the class RS0 = OEMS of linearly
ordered …nite pseudometric spaces. We may view RS0 as a class OM[S;q] of
lineraly ordered metric spaces with distances in the set S [ fqg. Note that
we have the inclusion OM[S;q] OMfS;qg . If we take K = OMS then by
De…nition 3 we obtain the class RS1 = CE[OMS ] of convex linearly ordered
pseudometric spaces with distances in the set S. We may view RS1 as the
subclass of OMfS;qg consisting of those (A; fRsA gs2Sq ; ) in RS1 such that
for every x, y, z 2 A we have
d(x; z) = q; x
y
z ) d(x; y) = d(y; z) = q:
When we want to view RS1 as a subclass of OMfS;qg then we will denote it
by CM[S;q] . More precisely sets of diameter q are intervals with respect to
. It is known that MS is not always a Fraïssé class, see [4]. In particular
for S = N or S = f1; :::; ng for some n 2 N , MS is a Fraïssé class. Therefore
we have the corresponding Fraïssé limits M[S;q] , OM[S;q] , CM[S;q] , MfS;qg
and OMfS;qg . It follows from [15] and [14] that MS1 is a Ramsey class for
S = N or S = f1; :::; ng.
Often M[S;q] is a proper subclass of MfS;qg , but there is an example when
these two classes are equal. Let dif(S) = inffs2 s1 : s1 2 S; s2 2 S; s1 <
s2 g. If 0 < q <dif(S) then we have M[S;q] = MfS;qg . Moreover, we have
two Fraïssé ordered expansions of the class MfS;qg : CM[S;q] and OMfS;qg .
The results of Sections 4 and 5 give us the following.
Corollary 5. Let S 6= ; be a countable subset of (0; +1) such that inf S >
0. Let q 2 R be such that 0 < q < inf S. Let MS be a Fraïssé class and let
MS1 be a Ramsey class. Then we have the following.
(i) OM[S;q] is not a Ramsey class, but CM[S;q] is a Ramsey class.
(ii) CM[S;q] satis…es OP with respect to M[S;q] , but OM[S;q] does not
satisfy OP with respect to M[S;q] .
If 0 < q <dif(S) then we have the following:
(iii) OMfS;qg is not a Ramsey class, but CM[S;q] is a Ramsey class whose
reduct is MfS;qg .
(iv) CM[S;q] satis…es OP with respect to MfS;qg , but OMfS;qg does not
satisfy OP with respect to MfS;qg .
We also have the following topological consequences of Theorem 20.
RELATIONAL QUOTIENTS
45
Corollary 6. Let S 6= ; be a countable subset of (0; +1) such that inf S >
0. Let q 2 R be such that 0 < q < inf S. Let MS be a Fraïssé class and let
MS1 be a Ramsey class. Then we have the following
(i) Aut(CM[S;q] ) is an extremely amenable group, but Aut(OM[S;q] ) is
not an extremely amenable group.
(ii) XCM[S;q] is the universal minimal Aut(M[S;q] )-‡ow, but XOM[S;q] is
not a minimal Aut(M[S;q] )-‡ow.
If 0 < q <dif(S) then we have the following:
(iii) Aut(OMfS;qg ) is not an extremely amenable group.
(iv) XCM[S;q] is the universal minimal Aut(MfS;qg )-‡ow, but XOMfS;qg is
not a minimal Aut(MfS;qg )-‡ow.
9. Finite ultrametric spaces
Let S 6= ; be a subset of (0; +1). We denote the class of …nite ultrametric
spaces with distances in the set S by US . As in the case of metric spaces,
we may view US as a class of …nite structures in the signature L = fRs gs2S ,
where each Rs is a binary relational symbol. To each ultrametric space
(A; d) we assign the structure (A; fRsA gs2S ) in the same way we assign such
a structure to each metric space. For each countable set S, US is a Fraïssé
class with limit US , see [13] and [14]. Note that the class MS is not
necessarily a Fraïssé class. Let A = (A; d) 2 US and let s 2 S. Then
open balls of radius s form a partition of the set A. We say that a linear
ordering
on a set A is convex if for every a 2 A and every s 2 S the
open ball with center a and radius s is an interval with respect to . We
denote the collection of all convex linear orderings on the ultrametric space
A by co(A). Adding convex linear orderings to …nite ultrametric spaces
with distances in the set S we obtain the class:
CU S = f(A; fRsA gs2S ; ) : (A; fRsA gs2S ) 2 US ; 2 co((A; fRsA gs2S ))g:
By De…nition 1 we obtain the class NS = EU S of …nite pseudoultametric
spaces with distances in the set S. De…nition 2 give us the class NS0 = OEU S
of linearly ordered …nite pseudoultrametric spaces with distances in the
set S. If we take K = CU S then by De…nition 3 we obtain the class
NS1 = CE[CU S ] of convex linearly ordered pseudoultrametric spaces. It is
known that US and CU S are Fraïssé classes, see [13] and [14], such that CU S
is a Ramsey class satisfying OP with respect to US . Lemma 1, Lemma 4
46
MIODRAG SOKIĆ
and Lemma 6 imply that NS , NS0 and NS1 are Fraïssé classes with limits
NS , NS0 and NS1 respectively. The results of Sections 4 and 5 give us the
following.
Corollary 7. (i) NS1 is a Ramsey class, but NS0 is not a Ramsey class.
(ii) NS1 satis…es OP with respect to NS , but NS0 does not satisfy OP with
respect to NS .
We also have the topological consequences of Theorem 20.
Corollary 8. (i) Aut(NS1 ) is an extremely amenable group, but Aut(NS0 )
is not an extremely amenable group.
(ii) XNS is the universal minimal Aut(N)-‡ow, but XNS0 is not a mini1
mal Aut(N)-‡ow.
We now give a new proof that CU S is a Ramsey class using our approach.
First, we note that it is enough to show that CU S is a Ramsey class for
…nite S. Suppose S = fs0 < s1 <
< sn g and S 0 = Snfs0 g. It would
be enough to show that the Ramsey property for CU S 0 implies the Ramsey
property for CU S . Note that structures from EU S 0 may also be viewed as
structures from US , every equivalence class being an open ball of radius s0 .
If we take K = CU S 0 then by De…nition 3 we obtain the class O EU S 0 of
convex linearly ordered pseudoultrametric spaces. Note that we may view
O EU S 0 as the class CU S . Then by the result of Section 4 we have that CU S
is a Ramsey class. It remains to note that for the set S with jSj = 1, the
Ramsey property of CU S follows from the classical Ramsey theorem.
Remark 4. If there is q such that 0 < q inf S, then taking T = S [ fqg,
the class EU S may be viewed as a class of ultrametric spaces UT .
10. Graphs
We denote the class of …nite graphs by G and its elements by (A; E) where
E is a binary relation on A that gives a graph structure. G is a Fraïssé class
with Fraïssé limit G, also know as the Random graph. Adding arbitrary
linear orderings to graphs we obtain the class
OG = f(A; E A ; ) : (A; E A ) 2 G; 2 lo(A)g:
OG is a Ramsey class, see [17] and [18]. By De…nition 1 we obtain the
class H = EG. De…nition 2 gives us the class H0 = OEG of linearly ordered
RELATIONAL QUOTIENTS
47
structures from H. If we take K = OG then by De…nition 3 we obtain the
class H1 = O EG of convex linearly ordered structures from H. Lemma 1,
Lemma 4 and Lemma 6 imply that H, H0 and H1 are Fraïssé classes with
limits H, H0 and H1 respectively. The results of Sections 4 and 5 give us
the following.
Corollary 9. (i) H1 is a Ramsey class, but H0 is not a Ramsey class.
(ii) H1 satis…es OP with respect to H, but H0 does not satisfy OP with
respect to H.
We also have the topological consequences of Theorem 20.
Corollary 10. (i) Aut(H1 ) is an extremely amenable group, but Aut(H0 )
is not an extremely amenable group.
(ii) XH1 is the universal minimal Aut(H)-‡ow, but XH0 is not a minimal
Aut(H)-‡ow.
Remark 5. The group Aut(G) of automorphisms of the random graph is
N
o Aut(G) where
a subgroup of Aut(H). Moreover we have Aut(H) = S1
S1 is the group of permutations of natural numbers. Note that if we drop
the equivalence relation from the structure H then the resulting structure is
not isomorphic to G because it does not satisfy the extension property of the
Random graph.
11. Chains
Let D =
1
S
i=1
fig
Q and let D = (D; ) be a poset such that for all (i; p)
and (j; q) from D we have
(i; p)
(j; q) , (i = j and p
q):
We denote by D the class of all …nite posets isomorphic to some subposet
of D. Let A = (A; ) be a poset in D, and let 2 lo(A). We say that is
convex on A if for all x; y; z 2 A we have
(x
y
z and x
z) ) x
y
z:
This means that
agrees on every maximal chain in A and makes every
maximal chain into an interval. We denote the collection of convex linear
orderings on A by co(A). Adding convex linear orderings to structures from
D we obtain the class
OD =f(A; ; ) : (A; ) 2 D; 2 lo((A; ))g:
48
MIODRAG SOKIĆ
It is shown in [23] that D is a Fraïssé class, and in [26] that OD is a Fraïssé
class with Ramsey property.
By De…nition 1 we obtain the class J = ED. De…nition 2 gives us the
class J0 = OED of linearly ordered structures from J . If we take K = OD
then by De…nition 3 we obtain the class J1 = CE[OD] of convex linearly
ordered structures from J . Lemma 1, Lemma 4 and Lemma 6 imply that
J , J0 and J1 are Fraïssé classes with limits J, J0 and J1 respectively. The
results of Sections 4 and 5 give us the following.
Corollary 11. (i) J1 is a Ramsey class, but J0 is not a Ramsey class.
(ii) J1 satis…es OP with respect to J , but J0 does not satisfy OP with
respect to J .
We also have the following topological consequences of Theorem 20.
Corollary 12. (i) Aut(J1 ) is an extremely amenable group, but Aut(J0 ) is
not an extremely amenable group.
(ii) XJ1 is the universal minimal Aut(J)-‡ow, but XJ0 is not a minimal
Aut(H)-‡ow.
N
o
Remark 6. We have the topological group isomorphism Aut(J) = S1
Aut(D) where S1 is the group of permutations of natural numbers. Note
that if we drop the equivalence relation from the structure J then the resulting
structure is not isomorphic to D because it does not satisfy the extension
property of the Fraïssé structure D.
12. Modifications
We discuss few variations in our approach.
(1) Let A = (A; fRiA gi2I ; E A ) be a structure from the class EK given by
A=E
De…nition 1. Then we have the structure A=E = (A=E; fRi gi2I )
from the class K. If ni is the arity of the relation Ri then for all
x1 ; :::; xni ; y1 ; :::; yni 2 A with xj E A yj , 1 j ni , we have
RiA (x1 ; :::; xni ) , RiA (y1 ; :::; yni ):
According to this de…nition of the relation RiA , it is not correct
to say that RiA does not take place inside E A -equivalence classes.
A=E
For example, if ni = 2 and Ri
is such that there is a 2 A=E
RELATIONAL QUOTIENTS
49
A=E
with Ri (a; a) then for x; y 2 A with [x]E A = [y]E A = a we have
RiA (x; y). We may avoid this by putting an additional requirement
RiA (x1 ; :::; xni ); [xi ]E A = [xj ]E A ) xi = xj :
Let A be the structure de…ned in this way. Then A and A have the
same automorphism group. Moreover, all statements in this paper
stay valid if we make a change of de…nition of relations in this way.
We may make similar change for classes given by De…nition 4.
(2) We would like to explain our use of equivalence relations. Suppose
that we have the class K of …nite relational structures in a signature
fRg where R is a binary symbol. Let EK be the class of structures
given by De…nition 1. We consider the structures A = (A; RA ; E A )
and B = (B; RB ; E B ) from EK such that A = fa1 ; a2 g, B = fb1 ; b2 g,
(a1 ; a2 ) 2
= RA , (b1 ; b2 ) 2
= RB , (a1 ; a2 ) 2
= E A , (b1 ; b2 ) 2 E B . Note
that A and B are not isomorphic. If we drop the equivalence relations from our structures we obtain structures A = (A; RA ) and
B = (B; RB ) which are isomorphic. Therefore, replacing points by a
collection of points without introducing equivalence relations would
make the consideration of Ramsey property much harder.
(3) There are special cases of structures for which we do not need to
use equivalence relations. Let K be the class of …nite relational
structures in a signature fRi gi2I with arity fni gi2I . Assume that
for every structure A = (A; fRiA gi2I ) from K, all i 2 I, and all
x1 ; :::; xni 2 A we have
RiA (x1 ; :::; xni ) ) jfx1 ; :::; xni gj = ni ;
and for every x; y 2 A there is i 2 I, and there are x1 ; :::; xni
such that
RiA (x1 ; :::; x; :::y; :::; xni 2 ):
2
2A
Let EK be the class given by De…nition 1, and let EK be the class
obtained by dropping the equivalence relations from the structures in
the class EK. Then we are able to recover the equivalence relation in
the structures from EK. For B = (B; fRiB gi2I ) in EK and x; y 2 B
we take (x; y) 2 E B i¤ there is no i 2 I, and x1 ; :::; xni 2 2 B such
that
RiA (x1 ; :::; x; :::y; :::; xni 2 ):
50
MIODRAG SOKIĆ
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Department of Mathematics, California Institute of Technology, Pasadena,
CA 91125
E-mail address: msokic@caltech.edu