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3.4.
CHARACTERIZATION FOR TRANSITIVITY
181
Now we assume that for some
u1, u2∈ ρ
, the set
ν(p)
Pp(u1, u2)
is innite. Then by compactness, for a realizationaofp, the set
q(a, y) {θ
(a, y)} ∪ {¬θv(a, y) | v ∈ Pp(u1, u2)}
u1,u
2
is consistent. Consider realizationsbandcofpsuch that
θ
(a, b) ∧ θ
u
1
and
(a, c) /∈ Ipby the construction ofq. Thus the relation
transitive and we obtain
3.4.0.2. Denition.
ministic
for any
3.4.0.3. Example.
ory
Th(pm(G1, G2, P))
and, thus, the structure
The determinacy of
and there are at most two points inP, or the group
is unit and either
almost determinacy of
3.4.0.4. Proposition.
P
is almost deterministic then
ν(p)
(b, c)
u
2
if the set
u1, u2∈ ρ
and
|= q(a, c)
. We have
(a, b) ∈ Ip,
(b, c) ∈ Ip,
(1) ⇒ (2).¤
bu1, u2c
.
ν(p)
A structure
is a singleton (is nonempty and nite)
P
is called (
ν(p)
almost)deter-
By the denition, any polygonometrical the-
(see [54]) has a unique1-type
P
is a monoid with non-negative labels.
ν(p)
P
means that the group
ν(p)
p(x) ∈ S(∅)
G1of sides is unit
G2of angles
P
contains unique line or
ν(p)
P
means that the group
ν(p)
If there is a model
P
ν(p)
G1is innite. The
G2is nite.
Mpand the structure
is a monoid.
Ipis not
|=
¤
Proof.
cle, for
u1, u2, u3, v,u1< 0,v < 0
no
v0∈ bu2, u3c
bu2, u3c
M(a)
Since the formula
there is a required label
Any deterministic structure
As noticed in Proposition 3.3.0.1, the unique obsta-
P
, where
to be a monoid, can be only the existence of labels
ν(p)
, for which
with
v ∈ bu1, v0c
consists of nitely many labels
|= p(a)
M(a) |= θ
, elements
(a, b) ∧ θ
u
1
θ
(b, y)
u2,u
3
u
2
is equivalentto the formula
v0= visuch that
v ∈ bu1, u2, u3c
. But, by the hypothesis, the set
v1, . . . , vk. Now we take in
b, c, d
(b, c) ∧ θ
such that
(c, d) ∧ θv(a, d).
u
3
M(a) |= θ
P
is a monoid (being almost
ν(p)
deterministic). It is generated by the monoid
where
bu, vc = {u ¯ v}
for
u, v ∈ ρ
ν(p)
.
and there are
0
(b, d).¤
v
0
P
ν(p)
= hρ
k
W
i=1
θ
v
ν(p)
(b, y)
i
; ¯i
,
,
182
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Thus, the deterministic monoids can be dened by usual Cay-
ley tables for monoids on a set of labels inUwhile the almost
deterministic monoids are represented by one-to-nite functions
with two arguments, i. e., by ternary predicates with nitely many
third coordinates for xed rst and second coordinates.
Considering deterministic structuresP, being restrictions of
the monoid
denote by
{u ¯ v}
P
0
P
for
u, v ∈ U0.
to some subalphabets
ν(p)
the
generating
monoid
U0of the alphabetU, we
hU0; ¯i
such that
bu, vc ∩U0=
The following proposition is a reformulation of Proposition
3.4.0.1.
3.4.0.5. Proposition.
T
having a model
Mp,
Let
p(x)
be a complete type of a theory
ν(p)
be a regular labelling function. The
following conditions are equivalent:
(1)
the relation
Ip(
on the set of all realizations ofpin a model
M |= T)is transitive;
(2)
the structure
P
is an almost deterministic monoid.
ν(p)
Note that there are no principal edges linking distinct realiza-
tions ofpif and only if the relation
is reexive, the denition of
ν(p)
Ipis antisymmetric. Since
and Propositions 3.1.0.6, 3.4.0.5
imply
3.4.0.6. Corollary.
ing a model
Mp,
Let
p(x)
be a complete type of a theoryThav-
ν(p)
be a regular labelling function. The following
conditions are equivalent:
(1)
the relation
Ip(
on the set of realizations ofpin any model
M |= T)is a partial order;
(2)
the structure
ρ
⊆ U≤0.
ν(p)
This partial order
P
is an almost deterministic monoid and
ν(p)
Ipis identical if and only if
ρ
ν(p)
= {0}
Ipis not identical, it has innite chains.
I
. If
p
Theorem 1.1.3.11 and Proposition 3.4.0.5 imply
3.4.0.7. Corollary.
T,ν(p)
be a regular labelling function. The following conditions
Let
p(x)
be a complete type of a small theory
are equivalent:
3.4.
CHARACTERIZATION FOR TRANSITIVITY
183
(1) Ip(
on the set of realizations ofpin any model
M |= T)is
an equivalence relation;
(2)
the structure
P
is an almost deterministic monoid and
ν(p)
there are no limit models overp;
(3)
the structure
P
is an almost deterministic monoid and
ν(p)
consists of non-negative labels.
In Corollary 3.4.0.7, the equivalence of (1) and (3) is implied
by the existence of
3.4.0.8. Denition.
terministic
if for any/some realizationaofpthe formula
Mpwithout the assumption of smallness ofT.
An element
u ∈ ρ
is called (
ν(p)
almost)de-
θu(a, y)
has a unique solution (has nitely many solutions).
Note that there is no negative almost deterministic element
for a theoryThaving an atomic model and nitely many nonprin-
cipal1-types in
elementuimplies that the type
tion
SIpis not symmetric, that is witnessed by the formula
Since for
some
k ∈ ω \ {0}
any realizationbof
mula
θu(b, y)
nonprincipal types we may assume that
each nonprincipal type
d
realizing the typep. By non-symmetry of the relation
formula
ϕ(x)∧θu(x, d)
S(T)
.3Indeed, otherwise the presence of a negative
p(x)
is nonprincipal and the rela-
|= p(a)
the isolating formula
, there exists a formula
ϕ(x)
there are exactlyksolutions of the for-
θu(a, y)
hasksolutions for
ϕ(x) ∈ p(x)
such that for
. Moreover, Moreover, since there are nitely many
ϕ(x)
is not consistent with
q(x) 6= p(x)
. Let
|= θu(a, d)
for someaand
has a solutioncwhich does not realizepand
θu(x, y)
SIpthe
any other nonprincipal1-type. Socrealizes some principal type,
which is isolated by a formula
solutions there is a formula
θu(c, y) ∧ µ(y) ` p(y)
. Then the formula
ψ(x)
µ(y)
. Since
θu(c, y)
such that
∃x(ψ(x) ∧ µ(y) ∧ θu(x, y))
has nitely many
|= θu(c, d) ∧ µ(d)
and
isolates the nonprincipal typep, for a contradiction.
At the same time, Example 1.4.1.7 illustrates that there are
theoriesTwith even deterministic negative elementsu, where there
are innitely many nonprincipal1-types in
S(T)
.
u
.
3
The
following arguments, in fact, are identical to the remark after the
proof of Proposition 1.4.1.6.
184
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.4.0.9. Proposition.
ministic then any element
Proof.
where
|= p(a)
Consider formulas
. Ifuandvare deterministic then all these for-
mulas have unique solutions, so the element
and the formulas
θ
u,v
If elementsuandvare(almost)deter-
v0in
(a, y)
and
u · vis(
θu(a, y),θv(a, y)
0
θ
(a, y)
v
almost)deterministic.
, and
v0∈ u · v
is unique,
are equivalent.
θ
Ifuandvare almost deterministic then the formulas
θv(a, y)
the set
formulas
, and
θ
(a, y)
u,v
u · v
is nite and there are nitely many solutions for the
0
θ
(a, y),v0∈ u · v.¤
v
have nitely many solutions. It implies that
Proposition 3.4.0.9 immediately implies
3.4.0.10. Corollary.
(
respectively
P
ν(p),ad
For any groupoid
)
to the set of(almost)deterministic elements
P
its restriction
ν(p)
is a monoid.
The following proposition presents a characterization for deter-
minacy of non-negative elements in
model
3.4.0.11. Proposition.
u ≥ 0inP
Mp.
If the model
is deterministic if and only if
ν(p)
P
assuming existence of the
ν(p)
Mpexists then an element
u−1· u = {0}
(a, y)
u,v
θu(a, y)
P
ν(p),d
.
,
,
Proof.
{b}
for some realizationsaandbofpin
{b}
, i. e.,
We assume now that
θu(a, y)
contrary that there are at least two solutions
we have
(b1≈ y)
formula
there is an isolating formula
θ
(b1, y)
u−1,u
Let an elementube deterministic, i. e.,
Mp. Then
u−1· u = {0}
, where
|= θ
, and
θ
u−1,u
|= p(a)
−1
(b1, a) ∧ θu(a, b2)
u
θ0(b1, y) ` θ
(b1, y)∧¬θ0(b1, y)
.
u−1· u = {0}
and prove that the formula
, has a unique solution. Assume on the
. Since
u−1,u
(b1, y)
then the consistency of the
and the existence of
θv(b1, y),v 6= 0
. It contradicts the condition
u−1· u = {0}.¤
θu(a, Mp) =
θ
(b, Mp) =
u−1,u
b1and
b2. Then
0 ∈ u−1· u,θ0(b1, y) =
Mpimply that
, such that
θv(b1, y) `
Unlike the determinacy there are no similar characterizations
for the almost determinacy.
3.4.
CHARACTERIZATION FOR TRANSITIVITY
185
3.4.0.12. Example.IfΓ = hM; Ri
is an acyclic undirected graph
consisting of vertices of xed positive degreeυthen for the unique
1-type
|= θn(a, b) ⇔ ρ(a, b) = n,n ∈ ω
the alphabetω, we have
n = n−1and
does not depend on
3.4.0.13. Proposition.
the structure
p(x) ∈ S(Th(Γ))
n · n = {0, 2n}
υ ∈ (ω ∪ {∞}) \ {0}.¤
0
P
is a group if and only if
ν(p)
, for the principal formulas
, and for the monoid
m · n = {m + n, |m − n|}
. At the same time the monoid
If
P
is a deterministic monoid then
ν(p)
ρ
ν(p)
θn(x, y)
, where
P
ν(p)
over
. In particular,
P
ν(p)
consists of non-
negative elements.
Proof.
negative then there are no labelsvsuch that
PFN(p) 6= ∅
Now we assume that
0
P
u ∈ ρ
is a group. Indeed, if
ν(p)
ν(p)
0 ∈ u · v
At rst we observe that, by denition, if
P
0
is not a group.
ν(p)
ρ
∩U−= ∅
ν(p)
and prove that the structure
PFN(p) = ∅
then for any element
then
, there is the (unique) inverse element
. As the monoid
P
is deterministic we obtain
ν(p)
u ∈ ρ
u ¯ v = 0
. Hence, if
v = u−1such that
u ¯ v =
ν(p)
is
0.¤
3.4.0.14. Corollary.
deterministic, and
If the model
0
P
is a group, then all elements in
ν(p)
Mpexists, the monoid
P
P
0
ν(p)
ν(p)
is
are
deterministic.
Proof.
non-negative elements then, as the monoid
by Proposition 3.4.0.11 each element in
3.4.0.15. Proposition.
Since by Proposition 3.4.0.13 the set
P
0
P
ν(p)
If the model
Mpexists then the set
of all non-negative deterministic elementsuin
ments
G
P
u · u−1= u−1· u = {0}
if
u−1are also deterministic, forms a deterministic submonoid
of the monoid
ν(p)
, and such that
ν(p)
Proof.
u, v ∈ ρ
≥0
ν(p),d
Since for any
then
P
ν(p),d
(G
ν(p)
belongs to
u · v
, consisting of deterministic elements of
)0is a group.
u ∈ ρ
≥0
ν(p),d
≥0
ρ
ν(p),d
the element
it suces to observe that
contains a unique element
element is deterministic by Proposition 3.4.0.9.
ρ
consists of
ν(p)
is deterministic,
ν(p)
is deterministic.
ρ
ρ
, for which ele-
ν(p)
u−1satisfying
v0and this
¤
¤
≥0
ν(p),d
186
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
P
ν(p)
•
@
¡
¡
¡
¡
@
¡
¡
@
•
•
•
@
@
©
@
P
ν(p),ad
@
P
ν(p),d
@
©
{0}
©
@
@
@
©
@
@
@
©
@
@
@
@
@
@
©
•
•
•
•
≥0
P
ν(p)
≥0
P
ν(p),ad
≥0
P
ν(p),d
G
ν(p)
¡
¡
¡
≤0
¡
P
•
ν(p)
¡
¡
¡
≤0
¡
P
•
ν(p),ad
¡
¡
¡
≤0
¡
P
•
ν(p),d
@
@
@
@
Figur
e 3.1
In Figure 3.1, a Hasse diagram is presented illustrating the links
of the structure
P
to subalphabets ofU. Here the superscripts
ν(p)
out on restrictions of
negative elements respectively, the subscripts
P
with structures above, being restrictions of
ν(p)
·≤0and
P
to the sets of non-positive and non-
ν(p)
·dand
·adindicate
·≥0point
the sets of deterministic and almost deterministic elements. By
Propositions 3.3.0.1 and 3.4.0.4, just
P
ν(p)
and
P
≤0
may not be
ν(p)
monoids.
The following proposition shows that for each label
the monoid
tion of a submonoid of
for which
P
ν(p)
u ∈ u · v
contains a monoid
P
and consisting of all labels
ν(p)
.
P
with0being a restric-
ν(p),u
u ∈ ρ
v ∈ ρ
ν(p)
ν(p)
,
3.5.
GRAPH AND MONOID COMPOSITIONS
187
3.4.0.16. Proposition.
for labels
Proof.
θw(x, y)
u, v, winP
Since
u ∈ (u·v)∩ (u ·w)
and a realizationaofp, there are realizations
ν(p)
If
P
, then
is a monoid and
ν(p)
u ∈ (u · (v · w))
, for formulas
u ∈ (u·v)∩(u·w)
.
θu(x, y),θv(x, y)
for which
|= θu(a, b) ∧ θu(a, c) ∧ θu(a, d) ∧ θv(b, c) ∧ θw(c, d).
Then
which
u ∈ bu, v, wc
. As
u ∈ u · v0. Hence
P
is a monoid, there is
ν(p)
u ∈ (u · (v · w)).¤
v0∈ (v · w)
3.5. Graph and monoid compositions
3.5.0.1. Denition.
graphs
where
Γ1= hX1; R1i
((a1, b1), (a2, b2)) ∈ R
conditions is met:
1)
(a1, a2) ∈ R1;
2)
a1= a2and
Similarly we dene the notion of monoid composition.
Let
S1and
and
S2⊆ U≥0. The
band
(see [20, 34])
hS1∪ S2; ¯i
v ¯ u = u
S2be monoids, for which0is the unit,
, where
for
u < 0
Recall [52] that the
and
Γ2= hX2; R2i
is the graph
if and only if some of the following
(b1, b2) ∈ R2.
composition
S1[S2]
of monoids
or the
S1and
hS1∪ S2; ¯i ¹ Si= Sifor
and
v > 0
.
composition
hX1× X2; Ri
sequentially-annihilating
S2is the algebra
i = 1,2
, and
b, c, dofp
for
Γ1[Γ2]
of
S1⊆ U≤0,
u ¯ v =
,
,
,
3.5.0.2. Proposition.
S1[S2]
S1and
is a monoid.
Proof.
We x a sequentially-annihilating band
S2are monoids it suces to check the associativity:
u2)·u3= u1·(u2·u3)
two of them belong to
[34]
. Any sequentially-annihilating band
for any three elements
U−or to
U+.
We check this property analyzing six cases:
(1) if
u1∈ U−and
u2, u3∈ U+then
(u1¯ u2) ¯ u3= u1¯ u3= u1= u1¯ (u2¯ u3);
S1[S2]
. Since
(u1·
u1, u2, u3, where exactly
188
Chapter
(2) if
u1, u3∈ U+and
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
u2∈ U−then
(u1¯ u2) ¯ u3= u2¯ u3= u2= u1¯ u2= u1¯ (u2¯ u3);
(3) if
u1, u2∈ U+and
u3∈ U−then
(u1¯ u2) ¯ u3= (u1¯ u2) ¯ u3= u3= u1¯ u3= u1¯ (u2¯ u3);
(4) if
u1, u2∈ U−and
u3∈ U+then
(u1¯ u2) ¯ u3= u1¯ u2= u1¯ (u2¯ u3) = u1¯ (u2¯ u3);
(5) if
u1, u3∈ U−and
u2∈ U+then
(u1¯ u2) ¯ u3= u1¯ u3= u1¯ (u2¯ u3) = u1¯ (u2¯ u3);
(6) if
u1∈ U+and
u2, u3∈ U−then
(u1¯ u2) ¯ u3= u2¯ u3= u1¯ (u2¯ u3) = u1¯ (u2¯ u3). ¤
3.5.0.3. Theorem.
For any group
hG; ∗i
, where the universe
consists of non-negative elements and0denotes the group unit,
and for the monoid
idempotent element−1, there is a theoryTwith a type
and a regular labelling function
coincides with the monoid
Proof.
Th(M)
ν(p)
with
We construct a structure
has a type
For this aim we consider the Ehrenfeucht's example
ck< c
k+1
,
k ∈ ω
antichain consisting of
polygon over the group
hG; Qgi
g∈G
, where
we replace each constant
elements of a copy ofG. Thus we form the composition
graphs expanded by relations
h{−1, 0}; +i
with the zero element0and the
p ∈ S(T )
ν(p)
h{−1, 0}; +i[hG; ∗i]
p(x) ∈ S(T )
0
P
= h{−1, 0}; +i[hG; ∗i].
ν(p)
and a regular labelling function
such that the monoid
.
M
such that its theory
hQ; <, cki
P
T =
k∈ω
0
ν(p)
, such that each elementais replaced by a<-
|G|
elements and forming a free1-generated
hG; ∗i
Qg= {(a, b) ∈ G2| a ∗ g = b},g ∈ G
ckby a unary predicate
isomorphic to the structure
Rkconsisting of
hQ; <i[G]
Rk,
k ∈ ω,(x < y),¬(x < y)∧¬(y <
G =
. Here
of
,
3.5.
GRAPH AND MONOID COMPOSITIONS
189
x),Qg,
set of formulas
p
the list of pairwise non-equivalent isolating formulas
ϕ(a, y) ` p(y)
g ∈ G
formula
non-negative labelsg. Since
<,g ∈ G
by the group
g ∈ G
. The unique nonprincipal1-type
∃y(Rk(y) ∧(y < x)),k ∈ ω
is exhausted by the formulas
. For any realizationaof
(a < y)
. We dene a regular labelling function
(a < y)
has the label
−1
and the formulas
< ◦ < = <,< ◦ Qg= Qg◦ < =
, and the links between elements of
hG; ∗i
, the monoid
0
P
coincides with the monoid
ν(p)
p(x)
is isolated by
ϕ(a, y)
and
ν(p)
such that the
Qg(a, y)
≥0
ρ
ν(p)
with
Qg(a, y)
have
are dened
h{−1, 0}; +i[hG; ∗i].¤
3.5.0.4. Theorem.
For any group
hG; ∗i
consisting of non-
negative elements with the unit element0and for the monoid
hω∗; +i
p ∈ S(T )
0
P
ν(p)
Th(M)
ν(p)
n ∈ ω
bolQ, and of binary predicate symbols
of non-positive integers, there exists a theoryTwith a type
and a regular labelling function
coincides with the monoid
Proof.
with
The language of
We construct a structure
has a type
0
P
ν(p)
p(x) ∈ S(T )
= hω∗; +i[hG; ∗i]
M
consists of unary predicate symbols
hω∗; +i[hG; ∗i]
and a regular labelling function
.
ν(p)
such that the monoid
.
M
such that its theory
(forming a coloring of the setM), of binary predicate sym-
Qg,
g ∈ G
.
We consider a connected acyclic directed graph
Γ = hM0; Qi
T =
Coln,
where each element has innitely many images and innitely many
preimages, i. e.,Γforms a free directed pseudoplane.
We dene an1-inessentialQ-ordered coloring
{∞}ofΓ
n},n ∈ ω
producing unary predicates
.
Coln= {a ∈ M0| Col(a) =
For the graphΓwe dene, by induction, relations
Q0 id
M
n+1
,
Q
0
Qn◦ Q,Q−n (Qn)−1,
Note that for the (unique) nonprincipal type
the set
andbofp, the pair
for some
n ∈ ω
is equivalent to the formula
{¬Colm(x) | m < ω}
(a, b)
n ∈ ω
.
of formulas, and for any realizations
is a principal arc if and only if
We assume that the formula
. Since for any
m, n ∈ ω
Q
Qn(x, y)
the formula
m+n
(x, y)
, then for theQ-structure
has the label
∃z(Qm(x, z) ∧ Qn(z, y))
Col:M0→ ω ∪
Qn,
n ∈ ω
.
p(x)
, isolated by
n ∈ Z
|= Qn(a, b)
−n ∈ U≤0,
,
,
:
a
190
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
on the set of realizations ofpthe structure
hω∗; +i
binary predicates
.
Now we consider the group
Qg,
g ∈ G
, by the rule:
hG; ∗i
and dene, on the setG,
P
0
coincides with
ν(p)
Qg= {(a, b) ∈ G2| a ∗ g = b}.
As in the proof of Theorem 3.5.0.3 the structure
forms a free1-generated polygon over the group
G = hG; Qgi
hG; ∗i
.
g∈G
We dene a model of a required theoryTas the composition
Γ[G]
of graphs with colored vertices and arcs such that each vertex
aofΓ
have the color
is replaced by a copy of structureG, for which all elements
Col(a)
. The relations
Qg, for
Γ[G]
, are composed
as the unions of corresponding relations in the copies ofG, and
the relationQ, in
b0∈ Cb,
vertices
(a, b) ∈ Q
a, b ∈ M0. The composition preserves the uniqueness of
the nonprincipal1-type
Γ[G]
, consists of all pairs
inΓ, and
p(x)
Ca,
.
(a0, b0)
, where
a0∈ Ca,
Cbare copies ofGreplacing
It remains to note that for any realizationaofpthe list of
pairwise non-equivalent isolating formulas
p(y)
is exhausted by the formulas
G
, and we have
P
0
¹ ω∗= hω∗; +i,P
ν(p)
Qn◦ Qg= Qg◦ Qn= Qnfor
n > 0,g ∈ G.¤
Qn(a, y),n ∈ ω,Qg(a, y),g ∈
ϕ(a, y)
0
¹ G = hG; ∗i
ν(p)
with
ϕ(a, y) `
and
3.6.
In this section, we collect basic structural properties of
I
-groupoids
P
ν(p)
groupoids and prove that any groupoidPsatisfying that list of
properties coincides with some groupoid
3.6.0.1. Denition.
sisting of a set
ments
, and zero0. As above we write
u ∈ U−,
{u} · {v}
considering an operation·on the set
A groupoid
U−of
u > 0
for any element
P = hP(U) \ {∅}; ·i
Let
U = U
negative elements
−
∪ {0}˙∪ U+be an alphabet con-
P
.
ν(p)
˙
, a set
u ∈ U+, and
u < 0
U+of
for any element
u · v
P(U) \ {∅}
is called anI-groupoid
positive ele-
instead of
.
if it
satises the following conditions:
-
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