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Файл:Classification of countable models of complete theories. Р.1. Monograph in two parts
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Index
of terms
Algebra
canonical, 166
Ershov, 214
of distributions of binary iso-
lating formulas, 165
of distributions of binary semi-
isolating formulas, 208
preordered with relative set-
theoretic operations and
the composition, 214
semi-associative, 179
Algebraic system, 6
Amalgam, 113, 117
I
-free, 158
free, 130, 158
Amalgamation, 113
Approach
semantic, 114
syntactic, 114
Arc, 66, 164
irreversible, 164
principal, 163
Associativity, 220
Atom, 214, 229
Axiom
coherence, 129
Band
sequentially-annihilating, 187
Basis, 156
everywhere extensible, 156
internal, 159
of generators, 156
Cardinalities uniformly bounded,
120
Cartesian product
of classes, 161
of diagrams, 161
Chain, 105
elementary, 37
over a type, 37
Class
amalgamation, 113
determines, 120
nitely generated over
121
(D0; 6 )
,
312
INDEX
OF TERMS
Frasse, 114
generative, 113, 116
coherent, 129
dominating, 138
hereditary, 135
minimal hereditary, 135
quantier free, 126
self-sucient, 129
smooth, 129
witnessing the nite clo-
sure property, 148
generic, 113, 116
complete, 137
having nite closures, 121
of diagrams
conal in a class of struc-
tures, 119
Coloring
ϕ
-ordered, 63
of structure, 60
n
-inessential, 63
almost inessential, 61
inessential, 61
innerly almost inessential,
61
innerly inessential, 61
of universe, 61
Combination
of structures, 56
almost inessential, 57
inessential, 57
over a structure, 56
of theories, 56
almost inessential, 57
conal in a structure, 119
transitive forking, 109
uniformly locally nite, 120
Classes
domination-equivalent, 139
Closure
algebraic, 94
denable, 94
intrinsic, 134
self-sucient, 134
transitive, 101
Color
of element, 60
inessential, 56, 57
of types
inessential, 56
Complement
of label, 241
Complement relative, 213
Component
connected
of graph, 67
generated by set, 144
of connectivity of graph, 67
Composition
of elements, 239
INDEX
OF TERMS
313
of graphs, 187
of labels, 213
of monoids, 187
Condition
of choice, 83
Cone, 94
Conjecture
Lachlan, 7
Pillay, 13
Vaught, 12
Zilber, 16
Conjunction
of labels, 213
Construction
Hrushovski, 16
Jonsson{Frasse, 16
Copiesc-isomorphic, 145
Copy of diagram, 116
Cycle, 67
self-sucient, 120
Digraph, 66
acyclic, 67
powerful, 94
prepowerful, 97
Disjoint union
of structures, 49
of theories, 49
Disjunction
of labels, 213
Edge, 164
Element
almost deterministic, 183, 202
deterministic, 183, 202
inducing the pairwise inter-
section property, 196
inverse, 191, 204, 230, 240
negative, 164, 190, 203, 206,
229, 237
Degree
of semi-isolation
of family of types, 219
of label, 218, 230, 240
of type pair, 219
Diagram, 115
deduced, 120
embeddable in a diagram, 119
strongly, 119
embeddable in a structure, 119
strongly, 119
neutral, 206, 229, 237
positive, 164, 190, 203, 206,
229, 237
Elimination of quantiers, 54
Embedding
of diagram in structure, 119
strong, 119
of diagram into diagram, 119
identical, 119
strong, 119
of set strong, 132
314
INDEX
OF TERMS
Estimate majorizing, 153
Example Peretyat'kin, 26
Examples Ehrenfeucht, 26
Expansion
of generative class, 148
Extension
forking, 103
non-forking, 103
Family
si
-minimal, 216
of sets
regular, 203, 237
regular, 165, 207
Formula
(p, q)
-preserving, 27, 163
(p, q)
-semi-isolating, 27
(q, p)
-principal, 34
Function
dimension, 129
labelling, 164, 206
regular, 165, 207
predimension, 129
prerank, 129
rank, 129
Ryll-Nardzewski, 13
semi-associative, 199
successor, 80, 172, 244
Fusion
Hrushovski, 150
of generative
classes, 149
of generic
structures, 149
theories, 149
k
-divides, 103
p
-preserving, 28
decomposition, 45
divides, 103
principal, 64
witnessing
isolation, 27
semi-isolation, 27
witnessing denability of set,
198
witnessing that the relation
SIpis non-symmetric, 30
Frasse limit, 127
Generation of closure operation,
150
Graph, 66
bipartite random, 108
connected, 67
directed, 66
Herwig, 151
undirected, 66
Group of automorphisms
transitive, 79, 94
Groupoid
generating the strict order prop-
erty, 198
INDEX
OF TERMS
315
of binary isolating formulas,
180
Intersection
of labels, 213
Isomorphism types
domination-equivalent, 36
Join
of
POSTC
-monoids, 233
free, 233
of groupoids
free, 200
Join of
groupoids, 200
Label, 164, 206
atomic, 214
distributive with relative com-
plements, 214
Lemma
amalgamation, 145
Length of route, 66
Limit
Frasse, 114
Hrushovski, 113
Limit generic, 121
Model
p
-limit, 40
q
-prime, 25
almost prime, 25
dominated, 33
limit
over a type, 40
not exceedqunder the Rudin{
lutions, 244
inverse, 207
invertible, 165, 207
isolating, 207
semi-isolating, 207
Labels
α
-almost identic, 217
∼α-equivalent, 217
consistent, 214
inconsistent, 214
reciprocally inverse, 165, 207
Lattice
Keisler preorder, 33
prime over a tuple, 25
Models
RK
-equivalent, 34
strongly, 34
domination-equivalent, 34
strongly, 34
limit equivalent, 44
Rudin{Keisler equivalent, 34
Monoid
(α, ω)
-deterministic, 222
(α, n)
-deterministic, 222
α
-deterministic, 222
316
PIP
-special, 197
INDEX
of theories, 56
OF TERMS
almostα-deterministic, 222
almost deterministic, 181
deterministic, 181
generating, 182
of binary semi-isolating for-
mulas, 221
preordered with relative set-
theoretic operations and
a composition, 228
special, 197
Monotony, 212, 223, 234
Morleyzation, 76
Neighbourhood of set in
S(A)
32
Neighbourhood of set in
Sn(A)
32
Neighbourhood of set in type,
32
Number iterative, 150
Path, 66
Point, limit, 60
Preorder
of semi-isolation, 29
Rudin{Keisler, 33
Principal of minimization of it-
erations, 153
Problem
Lachlan, 7, 14
for simple theories, 15
Spectrum, 6, 11
Vaught, 6
Product
,
Cartesian, 161
Property
,
c
-amalgamation, 145
d
-amalgamation, 117
d
-covering, 130
d
-uniqueness, 118
amalgamation, 113
Order
denable, 106
dense, 105
strict, 105
identical, 105
Pair (of diagrams) minimal, 131
Pairing
of structures, 56
over a structure, 56
consistent extension of chains
of models prime over tu-
ples, 47
nite closure, 147
joint embedding, 120
local realizability, 117
pairwise intersection, 94
global, 97
local, 97
INDEX
OF TERMS
317
uniformd-amalgamation, 140
Pseudoplane free directed, 68
Quantier reduction, 252
Quasi-neighbourhood of setBin
S(A)
, 31
Quasi-neighbourhood of setBin
Sn(A)
, 31
Quasi-neighbourhood of set in a
type, 31
Quotient-class, 160
Quotient-diagram, 160
Rank
of semi-isolation, 215
of family of types, 216
of label, 215, 230, 240
of type pair, 216
Reduction of quantiers, 54
Representative of type, 116
Route, 66
Self-suciency relation, 113
Semi-associativity
left, 175
right, 175
Sequence
dening, 59
independent, 104
irregular in a model, 59
regular in a model, 59
equivalent, 70
Set
a
-denable,
a
-form
k
-inconsistent, 103
78
ula, 107
closed, 131
formula denable, 198
independent, 104, 156
of arcs, 66
of inverse labels, 207
of vertices, 66
partially ordered directed
downward, 101
upward, 101
self-sucient, 119
Structure, 6
(α, ω)
-deterministic, 234
(α, n)
-deterministic, 234
(D0; 6)
(D0; 6)
(D0; 6)
(K0; 6)
α
o
-minimal, 49
-generic, 121
-homogeneous, 126
-universal, 126
-generic, 113
-deterministic, 234
almostα-deterministic, 234
almost deterministic, 181, 201
atomic, 214
colored, 60
deterministic, 181, 201
Sequences
318
INDEX
OF TERMS
nitely generated over
121
generic, 113
having nite closures, 121
locallyα-deterministic, 235
self-sucient, 113
weakly saturated, 55
Subalgebra
partial
(α, ω)
-deterministic, 222
(α, n)
-deterministic, 222
α
-deterministic, 222
almostα-deterministic, 222
Subdiagram
strong, 193
Subdiagram strong, 119
Subgroupoid
(D0; 6 )
,
Ryll-Nardzewski, 13
Theories
characteristically equivalent,
Theories similar, 97
Theory, 6
(n, p)
(D0; 6)
1
-locally countably categori-
∆
-based, 54
over a set, 54
λ
-stable, 58
o
-minimal, 49
almost∆-based, 54
almostω-categorical, 74
colored, 61
49
-invariant, 97
-generic, 126
cal, 90
deterministic, 191
Subset strong, 119
Substructure
deterministic, 205
strong, 113
System
algebraic, 6
System of closuresE-stepped spe-
cial, 154
with minimality condition, 153
Theorem
on existence and uniqueness
of generic structure, 122
Ehrenfeucht, 7, 13, 25
generic, 114, 126
having the innite weight, 104
having the strict order prop-
erty, 77
of acyclic pairing function, 94
reduced over a type, 75
repsesenting
POSTCR-structure,
247
simple, 103
small, 25
special, 196
strongly minimal, 244
INDEX
OF TERMS
319
supersimple, 103
transitive, 63
Tuple
dependent, 104
having colorn, 72
having innite color∞, 72
independent, 104
isolates a tuple, 27
semi-isolates a tuple, 27
Tuples
dependent, 104
over a set, 104
independent, 104
over a set, 104
Type
(p1, . . . , pn)
(p1, . . . , pn)
∆
-based, 55
k
-divides, 103
p
-principal, 75
-powerful, 91
-principal, 75
dened by a set of formulas,
54
divides, 103
isolated by a set of formulas,
54
not exceedqunder the Rudin{
Keisler preorder, 33
powerful, 25
internally, 91
Types
RK
-equivalent, 34
strongly, 34
domination-equivalent, 34
strongly, 34
realization-equivalent, 34
strongly, 34
Rudin{Keisler equivalent, 34
strongly, 34
Undigraph, 66
acyclic, 67
Union
of labels, 213
of structures
over a structure, 56
Universe, 115
dominated, 33
forks, 103
free, 95
good, 42
having the innite own weight,
104
Variety
of generative classes, 160
of generic algebras, 160
Vertex, 66
Witness for a property, 148
Index
(α, β)
a, b)
(
(≥ 0)
(µ+)0(p1, p
(a0, an)
(a0, an)
(n, p)
(p, ϕ, q)
of symbols
-restriction, 219
,
25
-associativity, 175
n+1
-path, 66
-route, 66
-type, 74
-arc, 163
principal, 163
(p, ϕ, q)
-edge, 163
principal, 164
(p, q)
-atom, 238
(p ↔ q)
(p → p)
(p → q)
-formula, 28, 163
-formula, 28
-formula, 27
(p1, . . . , pn)
(q ← p)
(D1; 61) F
(D1; 61) F
(K0; 6)
(F
-formula, 27
, 113
ESS)n
i=1
(D0;60)
(D0;60)
(Di; 6i)
)
, 249
-type, 74
ESS
(D2; 62)
(D2; 62)
, 155
, 155
, 149
A
-class, transitive forking, 109
a
,
A
25
a
,
A ∪
I
I(T, λ)
I
25
-groupoid, 190
, 25
M
, 29
p
IR, 167
IR-structure, 204
n
-absorbing, 248
almostn-absorbing, 248
I
-groupoid, 180
ν(p)
P (p1, X1, p2, X2, . . . , pk, Xk, p
165
P (p1, u1, p2, u2, . . . , pk, uk, p
166
Pp(X1, X2, . . . , Xk)
Pp(u1, u2, . . . , uk)
Q
-cone
, 167
, 167
lower, 94
upper, 94
Q
-image
k+1
k+1
)
,
)
,
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