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3.2.
EXAMPLES
171
has a covering element then the formula
the disjunction of consistent formulas
θ−1(a, y) ∧ ¬θ
mula
θ−1(a, y)
We consider, as a theory with
−1,−1
.
(a, y)
, but it is impossible for the principal for-
ρ
ν(p)
theoryT, i. e. the theory of a structure
structure
that
ck< x,k ∈ ω
θ−1(a, y) = (a < y)
IV. Let
lim
k→∞
hQ; <i
ck= ∞
by adding constants
. The type
p(x)
, isolated by the set of formulas
, has exactly two non-equivalent isolating formulas:
and
θ0(a, y) = (a ≈ y)
ρ
= {−1, 0, 1}
ν(p)
. Realizing this equality, we con-
θ−1(a, y)
θ−1(a, y) ∧ θ
= {−1, 0}
M
ck,
ck< c
, where
is equivalent to
−1,−1
(a, y)
and
, the Ehrenfeucht's
, formed from the
,
k+1
|= p(a)
k ∈ ω
, such
.
sider the Ehrenfeucht's example, where each elementais replaced
by an<-antichain consisting of two elements
|= θ1(a0, a00) ∧ θ1(a00, a0)
for the type
p(x)
. Then we have the following equalities
isolated by the set of formulas
Pp(−1, −1) = Pp(−1, 1) = Pp(1, −1) = {−1},Pp(1, 1) = {0}
V. The equality
ρ
= {−2, −1, 0}
ν(p)
a0and
with
Pp(−2, −2) = {−2}
a00such that
0
c
< x,k ∈ ω
k
.
and
Pp(−2, −1) = Pp(−1, −2) = Pp(−1, −1) = {−1}
can be fullled by two dense strict orders
all realizations of a nonprincipal type such that
<1and
<2on the set of
<1immerses
<2:
<1◦ <2= <2◦ <1= <1.
:
VI. Consider a dense linearly ordered set
Th(M)
ν(p)
and2to
ρ
ν(p)
icates
, and the unique1-typepofT. Dene a labelling function
, for which0corresponds to the formula
(y < x)
,
Pp(1, 1) = {1},Pp(2, 2) = {2}
VII. Take a group
Qg,
g ∈ G
. We have
ρ
= {0, 1, 2},Pp(1, 2) = Pp(2, 1) =
ν(p)
.
hG; ∗i
and dene on the setGbinary pred-
, by the following rule:
M = hQ; <i,T =
(x ≈ y),1to(x < y)
Qg= {(a, b) ∈ G2| a ∗ g = b}.
If
p(x)
is a type (of a theoryT) realized in any model
M |= T
containingGexactly by elements inGconnected by denable re-
lations
Qg, then the typepis isolated, the setGis nite, and
ρ
ν(p)
consists of non-negative elements bijective with elements inG. If
,
172
ρ
ν(p)
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
consists of non-negative elements, is bijective withG, and the
set of realizations of a principal typepis not xed, then, assuming
the smallness of the theory, the setGis innite and the number
of connected components with respect to the relation
Q
S
Q
g∈G
is not bounded. At last if the typepis not isolated then the num-
ber ofQ-components on sets of realizations ofpis also unbounded
although the setGcan be nite.
The Cayley table of the group
Pp(·, . . . , ·)
relations
on the set
Qg.
ρ
in accordance with links between the
ν(p)
hG; ∗i
denes operations
g
VIII. Applying to a concrete group we consider the structure
M hZ;s
s(n) = n + 1
theory
is exhausted by the list:
ω
.
The set
(1)
Th(M)
ρ
i
with the unary
for each
n ∈ Z
successor functions:
Z ↔ Z
. For the unique1-typepof the
, the set of pairwise non-equivalent formulas
(x)
and
y ≈ s . . . s
consists of non-negative elements linked by the
ν(p)
|
n
{z}
times
x ≈ s
|
n
. . . s
{z}
times
, where
θu(x, y)
(y),n ∈
additive group of integers.
IX. Consider the type
the formula
ϕ(x1, x2, y1, y2)
ϕ(x1, x2, y1, y2) ≡
where
θ0(x1, x2, y1, y2) = (x1≈ y1< y2≈ x2),θ−1(x1, x2, y1, y2) =
q(x, y)
dened by
−3
_
θi(x1, x2, y1, y2),
i=0
in Example 1.1.4.2. Taking
x1≤ y1< y2≤ x2we get
(x1< y1< y2≈ x2),θ−2(x1, x2, y1, y2) = (x1≈ y1< y2< x2)
θ−3(x1, x2, y1, y2) = (x1< y1< y2< x2)
table illustrates the algebra of isolating formulas for
Table 3.1
. The following Cayley
q(x, y)
:
,
P
0 −1 −2 −3
q
0 0 −1 −2 −3
−1 −1 −1 −3 −3
−2 −2 −3 −2 −3
−3 −3 −3 −3 −3
3.2.
EXAMPLES
173
X. The structure
M0= hM0; Qi
in Example 1.2.3.5, equipped
by1-inessentialQ-ordered coloring, denes unique nonprincipal
1
-type
generated freely by one negative label, say
ned by the monoid
the predicateQas a disjoint union of predicates
gebra of isolating formulas for the type
generators
p∞(x)
. The algebra of isolating formulas for that type is
−1
, and so it is de-
hω∗; +i
. Representing, in Example 1.2.3.5,
p∞(x)
Q0and
has two negative free
Q1the al-
u1, u2and it is dened by free monoid with these two
generators. If the predicateQis represented by a disjoint union of
predicates
M0, then the algebra of isolating formulas for the type
|I|
free generators.
Qi,
i ∈ I
, each of which denes a structure of the form
p∞(x)
has
XI. In Example 1.3.3.1, the algebra of isolating formulas for the
type
p∞(x)
predicate
has three generating labels: negative labelsuandvfor
Q
and
R2, respectively, and a positive labelwfor the
R1. The labelsuandvgenerate the free subalgebra with
two-generated monoid. The free subalgebra, including the group
Z
, is generated by the labelw. Here we have the following equality
for the link between the labelsuandw:
P
(u, ww . . . w) = {u}
p
∞
.
XII. Consider an arbitraryλ-cubeC[54, Section 2.15], [421].
It is known that all isolating formulas
C
, are represented by
andbfor
|= dk(a, b)
dk(a, y)
, wherekis the distance between
. Assuming that each labeluis denoted by
θu(a, y)
, linking elements in
a natural number, dening that distance, for the unique1-type
and labels
m, n ∈ ω
the set
Pp(m, n)
consists of all numbers
a
p
n
X
|m +
i=1
where each
δiis equal to0or1. If the cardinalityλis nite then
we choose only numbers that do not exceed
XIII. If a structureAconsists of distinct constants
(−1)
δ
i
|,
2λ.
ci,
and does not have other language symbols, then for1-types
containing formulas
ci) | i ∈ I}
, if the setIis innite, the algebra of isolating formulas
(x ≈ ci)
, and for the type
p∞(x) = {¬(x ≈
consists of the following labels:
i ∈ I
pi(x)
,
,
174
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
1)0for the formula
type
piwith itself the label0is denoted by
realizations of
(if
|I| ≥ ω
2)
(i, j)
ciof
piwith the realization
3)
(∞, j)
of
p∞with the realization
p∞with themselves the label0is denoted by
);
for the formula
for the formula
(x ≈ y)
; linking the realization of the
(x ≈ ci)∧(y ≈ cj)
cjof
pj,
i 6= j,i, j ∈ I
(x ≈ x)∧ (y ≈ cj)
cjof
pj,
j ∈ I
(i, i),i ∈ I
, and linking
linking the realization
;
linking realizations
(if
|I| ≥ ω
).
For the algebraAof distributions for binary isolating formulas,
we have
placing in the left side of this equality some types
p
i
l
p
i
j
Replacing in the left side of the equality some types
the right side is nonempty (and equal to
only if the type
(i1, i2), . . . , (i
replaced by
P (p
, (i1, i2), p
i
1
the right side is not empty (and equal to
, . . . , p
i
2
, (ik, i
i
k
k+1
), p
) = {(i1, i
i
k+1
p
by other types
i
j
{0}
) only if all types
k+1
coincide and are replaced simultaneously by coinciding types.
p
i
j
{(∞, ik)}or{(∞, ∞)}
p∞replaces types
, is)
s−1
(∞, i
are replaced by0, and the label
)
.
s+1
p
i
1
, . . . , p
i
s
,
s ≤ k
, the labels
(is, i
IfIis nite, the algebraAis preserved under expansions of
preserving its universeA.
(∞, ∞)
)}
. Re-
by
p∞,
)
s+1
A
)
is
3.3. Algebra of distributions for binary isolat-
ing formulas on the set of realizations of
a type
We consider a complete theoryT, a type
labelling function
sets
Pp(u1, . . . , uk),u1, . . . , uk∈ ρ
ν(p):PF(p)/PE(p) → U
ν(p)
isolating formulas.
We denote by
Mpand by
M(a)
an atomic model over a real-
izationaofp.
Below we prove some basic properties for sets
bu1, . . . , ukc Pp(u1, . . . , uk).
3.3.0.1. Proposition.
1.
A set
bu1, u2c
if for a realizationaofpand for some formula
p(x) ∈ S(T )
, a regular
, and a family of
,
k ∈ ω
, of labels for binary
is nonempty if and only
θv(x, y),θv(a, y) `
3.3.
ALGEBRA OF DISTRIBUTIONS FOR BINARY FORMULAS
θ
(a, y)
u1,u
2
2.
any
u1, u2∈ ρ
3.
tionaofpand for some formula
holds.
If a model
ν(p)
The set
bu1, u2, u3c
Mpexists then the set
.
is nonempty if and only if for a realiza-
θv(x, y),θv(a, y) ` θ
bu1, u2c
holds.
4.
For any
u1, u2, u3∈ ρ
the following inclusions are satis-
ν(p)
ed:
bbu1, u2c, u3c ⊆ bu1, u2, u3c,
bu1, bu2, u3cc ⊆ bu1, u2, u3c.
5.
For any
u1, u2, u3∈ ρ
, the inclusion
ν(p)
bu1, u2, u3c ⊆ bbu1, u2c, u3c
175
is nonempty for
u1,u2,u
(a, y)
3
holds if and only if for any
such that
v ∈ bv0, u3c
.
6. (Left semi-associativity)
u1, u2, u3∈ ρ
ν(p)
,
v ∈ bu1, u2, u3c
If a model
bbu1, u2c, u3c = bu1, u2, u3c.
7.
For any
u1, u2, u3∈ ρ
the inclusion
ν(p)
bu1, u2, u3c ⊆ bu1, bu2, u3cc
is true if and only if for any
such that
v ∈ bu1, v0c
.
v ∈ bu1, u2, u3c
8. (Criterion for right semi-associativity)
exists, where
|= p(a)
, then for any
u1, u2, u3∈ ρ
bu1, bu2, u3cc = bu1, u2, u3c
holds if and only if for any
θ
(y1, y) ∧ θv(a, y)
u2,u
3
9. (
(≥ 0)
p(a)
, then for any
-associativity)
u1, u2, u3∈ ρ
v ∈ bu1, u2, u3c
is realized in
If the model
ν(p)
M(a)
, where
there is
v0∈ bu1, u2c
Mpexists then for any
there is
If the model
ν(p)
the formula
by a principal arc
M(a)
u1≥ 0
v0∈ bu2, u3c
the equality
θ
(a, y1) ∧
u
1
exists, where
,
M(a)
(b, c)
.
|=
bbu1, u2c, u3c = bu1, u2, u3c = bu1, bu2, u3cc.
176
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Proof.
1, 2, 3, 5, 7 follow immediately by the denition. In
view of 4, 8 is an easy reformulation of 7.
4. For the proof of
arbitrary element
v0∈ bu1, u2c
, and for any realizationaofpwe have
v ∈ bbu1, u2c, u3c
bbu1, u2c, u3c ⊆ bu1, u2, u3c
. Then
0
θ
(a, x2) ` θ
v
θv(a, y) ` θ
u1,u
v0,u
(a, x2),
2
(a, y).
3
By (3.1), we obtain
θ
(a, y) ` θ
v0,u
3
u1,u2,u
3
Thus, (3.2) and (3.3) imply
θv(a, y) ` θ
and, consequently,
v ∈ bu1, u2, u3c
Now we prove the inclusion
an arbitrary element
v0∈ bu2, u3c
, and for any realizationaofpwe have
v ∈ bu1, bu2, u3cc
u1,u2,u
bu1, bu2, u3cc ⊆ bu1, u2, u3c
.
(a, y),
3
. Then
v ∈ bv0, u3c
(a, y).
v ∈ bu1, v0c
, we take an
for some
(3.1)
(3.2)
(3.3)
. Take
for some
0
θ
(a, y) ` θ
v
θv(a, y) ` θ
u2,u
u1,v
(a, y),
3
0
(a, y).
By (3.4), we obtain
0
θ
(a, y) ` θ
u1,v
u1,u2,u
(a, y).
3
Thus, (3.5) and (3.6) imply
and, consequently,
θv(a, y) ` θ
v ∈ bu1, u2, u3c
u1,u2,u
(a, y),
3
.
6. Take a realizationaofpand an element
Then for the principal formula
θ
u1,u2,u
(a, y)
3
and so
M(a) |= θ
(a, b1) ∧ θ
u
1
u
2
θv(a, y)
(b1, b2) ∧ θ
v ∈ bu1, u2, u3c
, we have
(b2, c) ∧ θv(a, c)
u
3
(3.4)
(3.5)
(3.6)
.
θv(a, y) `
3.3.
ALGEBRA OF DISTRIBUTIONS FOR BINARY FORMULAS
177
for some realizations
is atomic overawe have
0
θ
(a, b2)
v
hence
for some
v ∈ bv0, u3c
bitrarily, we obtain, by 5,
by 4, the equality
b1, b2, andcofp. Since the model
0
θ
(a, x2) ` θ
v
v0∈ bu1, u2c
. Since the element
bu1, u2, u3c ⊆ bbu1, u2c, u3c
bbu1, u2c, u3c = bu1, u2, u3c
9. By 4 and 6, it suces to prove
for any
element in
M(a)
Since the type
some label
bu1, bu2, u3cc
following arguments. Since
u1, u2, u3∈ ρ
bu1, u2, u3c
, there are realizations
M(a) |= θ
−1
(a, b1) ∧ θ
u
1
tp(c/a)
v0. As
v0∈ bu2, u3c
.
If
u2≥ 0
and
u3≥ 0
, where
ν(p)
. Since
u1≥ 0
b1, b2, cofp
u
2
is principal, we have
we also have the required inclusion by the
v ≥ 0
there is a non-negative element
−1
−1
we have
we obtain
v−1∈ bbu
, u
3
v ∈ bu1, bu2, u3cc.¤
−1
c, u
2
1
M(a)
. Then
v ∈ bu1, u2, u3c
(a, x2)
u1,u
2
θv(a, y) ` θ
and
M(a) |=
(a, y)
v0,u
3
is chosen ar-
that implies,
.
bu1, u2, u3c ⊆ bu1, bu2, u3cc
u1≥ 0
. Letvbe an arbitrary
there is the label
−1
u
1
and, in
such that
(a, b2) ∧ θ
and
v ∈ bu1, v0c
(a, c) ∧ θv(b1, c).
u2,u
3
M(a) |= θ
0
(a, c)
v
we obtain
by Proposition 3.1.0.6 (2), and
−1
−1
v−1∈ bu
c
. Applying Proposition 3.1.0.6 (3),
, u
3
−1
, u
c
2
, then, by 6,
1
and
for
v ∈
Proposition 3.3.0.1 implies
3.3.0.2. Corollary.
If there is a model
the following conditions hold:
1.
For any
u1, u2, u3∈ ρ
ν(p)
bbu1, u2c, u3c = bu1, u2, u3c ⊇ bu1, bu2, u3cc
are satised.
2. (Criterion of associativity)
equality
bbu1, u2c, u3c = bu1, bu2, u3cc
hold if and only if
θ
(a, y1) ∧ θ
u
arc
1
(b, c)
u2,u
.
u1≥ 0
(y1, y) ∧ θv(a, y)
3
or, for any
M(a)
, the equalities
For any
v ∈ bu1, u2, u3c
is realized in
, where
|= p(a)
u1, u2, u3∈ ρ
, the formula
M(a)
by a principal
ν(p)
, then
, the
178
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Note that if
semi-associativities) can be failed. For instance, if
then
bbu1, u2c, u3c
Mpdoes not exist the associativity (as well as
bu1, u2c = ∅
is also empty although
bu1, bu2, u3cc 6= ∅
is
admissible.
By Proposition 3.3.0.1, having
failed only by some labels
u1, u2, u3with
3.1.0.6 (1), in this case any label
Mpthe associativity can be
u1< 0
v ∈ bu1, u2, u3c
. By Proposition
is also negative.
The mechanism presented in the following example shows that the
fault of right semi-associativity is admitted for any distribution of
signs for nonzero labels
3.3.0.3. Example.
and a label
v ∈ bu1, u2, u3c \ bu1, bu2, u3cc
3.3.0.1 (8), for the non-realizability of the formula
θ
(a, y1)∧θ
u
1
(y1, y2)∧θv(a, y2)
u2,u
3
schema of the realization of a non-p-principal
u2, u3: there are small theories with
bu1, u2, u3c 6= bu1, bu2, u3cc.
Obtaining (3.7) with
u1< 0,u2, u36= 0
(i. e., by Proposition
by principal arcs) we consider the
(2, p)
(3.7)
ϕ(a, y1, y2)
-type in a model
Mpof small theory presented in Example 1.3.3.1. Dening the type
p(x)
we introduce a
and
Qvcorresponding to the labels
ω ∪ {∞}
of some graphΓproducing unary predicates
M0| Col(a) = n},n ∈ ω
(a) for any
a, b ∈ M0for which
(b) if
which
m < n < ω
|= Colm(c) ∧ Coln(d) ∧ Qα(d, c)
Q
- and
u
1
Qv-ordered
(for binary predicates
u1andv) coloring
, such that:
m ≤ n < ω
and
α = u1, v
|= Colm(a) ∧ Coln(b) ∧ Qα(a, b)
then there are no elements
.
Q
u
Col:M0→
Coln= {a ∈
, there are elements
;
c, d ∈ M0for
Moreover, using a generic construction forΓwe obtain the
unique nonprincipal1-type
{¬Coln(x) | n ∈ ω }
For each label
a binary predicate
.
ui,
i ∈ {2, 3}
Q
u
i
uiis positive, and with the
we introduce labels
0
v
,
n
and positive otherwise, such that
dene pairwise disjoint predicates
same color if
0
v
> 0
n
, and linking with the
p(x)
and it is isolated by the set
, depending on its label, we dene
linking only elements of the same color if
Q
n ∈ ω
-ordering of
u
i
, being negative if
bu2, u3c = {v
0
Q
linking only elements of the
v
n
Colifui< 0
u2< 0oru3< 0
0
| n ∈ ω}
n
0
Q
-ordering of
v
n
. Now
. We
Col
,
1
3.3.
ALGEBRA OF DISTRIBUTIONS FOR BINARY FORMULAS
0
if
v
< 0
. Moreover, we require the following condition: for any
n
element
principal
the formula
formula witnesses that the non-p-principal
akof the colorkthe formula
0
Q
-arcs exactly with
v
n
ϕ(a, y1, y2)
is not realized by principal arcs, since this
n ≥ k
ϕ(ak, y1, y2)
. It means that for
(2, p)
-type
179
is realized by
|= p(a)
,
q(y1, y2) p(y1) ∪ p(y2) ∪ {θ
is realized in
If the model
Mp.
¤
Mpexists then, using the left semi-associativity, by
(y1, y2)} ∪ {¬θ
u2,u
3
0
(y1, y2) | n ∈ ω }
v
n
induction on the number of brackets one prove that all operations
b·, ·, . . . , ·c
binary operation
semi-associativity, the values
ρ
, do not depend on sequences of placements of brackets for
ν(p)
where
Thus, having
being a (left)
erations
eration
of
u · v
u1u2. . . uk, the following distribution of parentheses:
. . .) · uk)
Since by the choice of the label0for the formula
equalities
the groupoid
acting on sets in
b·, ·c
on the set
P(ρ
ν(p)
P(ρ
) \ {∅}
)\{∅}
ν(p)
are generated by the
. If we have the right
bX1, X2, . . . , Xkc,X1, X2, . . . , Xk⊆
X
i,i+1,...,i+m+n
X
1,2,...,k
semi-associative
b·, ·, . . . , ·c
b·, ·c
will be also denoted by·and we shall writeuvinstead
= bX1, X2, . . . , Xkc
bX
i,i+1,...,i+m
, X
.
Mp, the groupoid
P
ν(p)
algebra, admits to represent all op-
by terms of the language
i+m+1,i+m+2,...,i+m+n
hP(ρ
b·, ·c
) \ {∅}; b·, ·ci
ν(p)
. Below the op-
c,
. If the right semi-associativity fails we shall suppose, for
(((u1· u2) ·
.
(x ≈ y )
X · {0} = X
P
ν(p)
and
{0} · X = X
has the unit
are true for any
{0}
, and it is a monoid if the
X ⊆ ρ
ν(p)
the
algebra is right semi-associative. We have
,
,
Y · Z =[{yz | y ∈ Y, z ∈ Z}
for any sets
Y, Z ∈ P(ρ
ν(p)
) \ {∅}
in this structure.
Thus the following proposition holds.
3.3.0.4. Proposition.
S1(T )
ν(p)
having the model
, any operation
For any complete theoryT, any type
Mp, and the regular labelling function
Pp(·, ·, . . . , ·)
on the set
interpretable by a term of the groupoid
P
ν(p)
P(ρ
.
ν(p)
) \ {∅}
p ∈
is
180
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.3.0.5. Denition.
of binary isolating formulas over the labelling function
I
-groupoid
ν(p)
.
The groupoid
P
is called the
ν(p)
groupoid
ν(p)
or the
Propositions 3.1.0.6 and 3.3.0.1 imply
3.3.0.6. Proposition.
S1(T )
having the model
the restriction of the groupoid
For any complete theoryT, any type
Mp, and the regular labelling function
P
to the set of non-positive(re-
ν(p)
p ∈
ν(p)
spectively non-negative)labels is a semi-associative subalgebra of
P
with the unit
ν(p)
{0}(and, moreover, it is a monoid).
3.4. Characterization for transitivity of the re-
lation
ministic
The following assertion gives a characterization of transitivity
of the relation
1
-typepalthough the proof implies the validity for any complete
typerof a theory with a model
Ip. Deterministic, almost deter-
I
-groupoids and elements
ν(p)
Ip. For simplicity we formulate and prove it for a
Mr.
,
3.4.0.1. Proposition.
theoryThaving a model
Let
p(x)
Mp, and
be a complete type of a complete
ν(p)
be a regular labelling func-
tion. The following conditions are equivalent:
(1)
the relation
Ip(
on the set of all realizations ofpin a model
M |= T)is transitive;
(2)
for any labels
Proof.
Let
(b, c) ∈ Ipwitnessed by isolating formulas
If the set
Pp(u1, u2)
by existence of
k
formula
hence
W
i=1
|= θ
θ
(a, c)
v
i
(a, y)
v
i
by the formula
a, b, c,(2) ⇒ (1)
a, b, c
u1, u2∈ ρ
be realizations ofpsuch that
, the set
ν(p)
is nite and consists of labels
Mp, the formula
. Since
|= θ
for somei. Thus,
θ
(x, y)
v
i
. In view of arbitrary choice of elements
u1,u
θ
u1,u
2
(a, c)
2
(a, c) ∈ Ipand it is witnessed
is true.
Pp(u1, u2)
θ
(a, y)
we have
is nite.
(a, b) ∈ Ipand
u
1
(a, y)
and
θ
u
2
(b, y)
v1, . . . , vkthen,
is equivalent to the
k
W
|=
θ
(a, c)
v
i
i=1
.
and
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