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Файл:Classification of countable models of complete theories. Р.1. Monograph in two parts
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INTR
ODUCTION
AND HISTORICAL SURVEY
As it was already mentioned in the preface, one of the main aim
of the modern model theory is a solution of
the problem of the description of functions
of pairwise non-isomorphic models for theoriesTin cardinalities
for various classes of theoriesT. An interest to this problem is
caused mainly by that the substantial structural theory is required
for its solution.
The problem of description of spectrum functions and
of classes of theories depending on these functions has paid and
continues to pay attention of wide group of model-theoretic spe-
cialists, forming an extensive area of researches. It is contained
in many works, among which we mention the following: books and
dissertations by U. Andrews [1]; J. T. Baldwin [2, 3]; O. V. Bele-
gradek [4]; S. Buechler [7]; C. C. Chang, H. J. Keisler [8]; Yu. L. Er-
shov, S. S. Goncharov [14]; Yu. L. Ershov, E. A. Palyutin [15];
Handbook of mathematical logic [23]; W. Hodges [28]; A. Pillay [39,
40]; B. P. Poizat [43]; G. E. Sacks [45]; S. Shelah [47]; F. O. Wag-
ner [59]; papers by U. Andrews [65]{[68]; B. S. Baizhanov [82, 83];
J. T. Baldwin, A. H. Lachlan [89]; M. Benda [125]; E. Bouscaren,
D. Lascar [131]; E. Bouscaren [132]; S. Buechler [134]; S. S. Gon-
charov, M. Pourmahdian [187]; B. Hart, S. S. Starchenko, M. Vale-
riote [195]; B. Hart, E. Hrushovski, M. S. Laskowski [196]; B. Her-
wig, J. Loveys, A. Pillay, P. Tanovic, F. O. Wagner [390]; B. Her-
wig [206]; E. Hrushovski [222]; K. Ikeda, A. Pillay, A. Tsuboi [229];
H. J. Keisler, M. Morley [253]; B. Khusainov, A. Nies, R. A. Shore
[254]; B. Kim [257]; K. Zh. Kudaibergenov [266, 267]; A. H. Lach-
lan [278]; D. Lascar [283]; J. Loveys, P. Tanovic [289]; L. F. Low,
Spectrum Problem
I(T, λ)
for the numbers
, i. e.,
λ
12
INTR
ODUCTION AND HISTORICAL SURVEY
A. Pillay [290]; L. Mayer [299]; T. S. Millar [303]{[307]; M. Morley
[311, 312]; A. S. Morozov [314]; T. G. Mustan [316]; B. Omarov
[324]; E. A. Palyutin [329]; E. A. Palyutin, S. S. Starchenko [331];
M. G. Peretyat'kin [335, 336]; A. Pillay [337]{[341], [344, 346];
R. Reed [364]; A. N. Ryaskin [367]; C. Ryll-Nardzewski [368];
J. Sae [370, 371]; S. Shelah [375]; S. Shelah, L. Harring-
ton, M. Makkai [377]; P. Tanovic [55], [448]{[452]; S. Thomas
[459]; A. Tsuboi [460]{[463]; R. Vaught [465]; R. E. Woodrow
[60, 475, 476].
As already mentioned (see [47, 196]), the spectrum problem is
solved as a whole for countable complete theories in uncountable
cardinalitiesλ.
The problem of the description of number
I(T, ω)
of pairwise
non-isomorphic countable models of the theoryTfor the given
classes of complete theories is not well-studied till present. Here,
it should to mention the
there does not exist a theoryTsuch that
Vaught Conjecture
ω < I(T , ω) < 2ω. This
, according to which
conjecture has been conrmed for theories of trees (J. Steel [391]),
unars (L. Marcus [296], A. Miller [309]), totally transcendental
modules (S. Garavaglia [176]), varieties (B. Hart, S. S. Starchenko,
M. Valeriote [195]), foro-minimal theories (L. Mayer [299]), for
theories of modules over some rings (V. A. Puninskaya [359,
360]; V. A. Puninskaya, C. Toalori [361]). In the class of stable
theories, the Vaught conjecture has proved forω-stable theories
(S. Shelah [375]; S. Shelah, L. Harrington, M. Makkai [377]), for
various classes of stable theories (E. R. Baisalov [80, 81];
S. Buechler [134, 135]; L. F. Low, A. Pillay [290]; L. Newelski [318]{
[323]), and for1-based theories with a nonisolated type over a nite
set such that this type is orthogonal to the empty set (P. Tanovic
[449]). P. Tanovic [454] proved that ifTis a superstable theory
with
U(T ) ≥ ωωand the generic type of any simple group denable
in
TeqofU-rank smaller than
then
I(T, ω) = 2ω. Structural properties related to the Vaught's
ωωis eventually strongly nonisolated
conjecture are investigated by B. S. Baizhanov, N. S. Tazabekova,
A. D. Yershigeshova, T. S. Zambarnaya [88].
Attempts of constructions of examples denying the Vaught con-
jecture had taken (see R. W. Knight [260]). However the problem
is still open.
INTR
ODUCTION AND HISTORICAL SURVEY
13
Another interesting conjecture is the
serts that for any countable theoryT, the condition
implies
I(T, ω) ≥ ω
. A. Pillay [340] has proven this conjecture for
stable theories and has shown (see [337]) that
I(T, ω) ≥ 4
. Finally that Conjecture has proven by P. Tanovic
Pillay Conjecture
dcl(∅) |= T
dcl(∅) |= T
. It as-
implies
[450, 453].
In 1959, independently C. Ryll-Nardzewski [368], L. Svenonius
[446], and E. Engeler [163] have published well-known theorem
representing a syntactic criterion for countable categoricity of a
theory (i. e., the conditions
I(T, ω) = 1
), according which the
countable categoricity of a theory is equivalent to a niteness of
number ofn-types of the theory for every naturalnand xed
set of free variables. It means that every countably categorical
theory is dened by one characteristic, namely, the
function
, that puts in the correspondence to each naturalnthe
Ryll-Nardzewski
number of types ofnxed free variables.
There are papers investigating properties of the Ryll-Nardzewski
functions. Among these papers we note the following: J. Schmerl
[372]; U. Andrews, A. M. Kach [69].
Many results are connected with
theories having nitely many
(> 1)
Ehrenfeucht theories
, i. e.,
countable models. R. Vaught
[465] proved that there are no complete theories having exactly
two countable models. Using the theory of the dense linear order,
A. Ehrenfeucht [465] has constructed initial examples of theories
havingncountable models for each natural
n ≥ 3
. Further re-
searches have been connected with constructions of Ehrenfeucht
theories possessing various additional properties, with ndings and
investigations of structural properties of Ehrenfeucht theories, and
also with ndings of classes of complete theories that do not con-
tain Ehrenfeucht theories.
M. G. Peretyat'kin [335] has constructed, for every
n ≥ 3
a complete decidable theory having exactlyncountable models
such that the unique is constructible. G. E. Sacks [369] proved that
every countable model of an Ehrenfeucht theory has a hyperarith-
metic representation. B. Omarov [324], M. G. Peretyat'kin [336],
T. S. Millar [303, 307], S. Thomas [459], R. E. Woodrow [476] have
constructed examples of Ehrenfeucht theories admitting constant
expansions up to theories with innitely many countable models,
,
14
INTR
ODUCTION AND HISTORICAL SURVEY
and also of non-Ehrenfeucht theories such that some theirs constant
expansions are Ehrenfeucht. R. E. Woodrow [475] is shown that
assuming the quantier elimination and restricting the language
on a binary predicate symbol and constant symbols, the count-
able complete theories with exactly three countable models are, in
essence, the Ehrenfeucht examples. A. Pillay [339] has shown that
an innite dense partial order is interpretable in any Ehrenfeucht
theory with few links. S. S. Goncharov and M. Pourmahdian [187]
have proved that every Ehrenfeucht theory has a nite Scott rank
[374]. It is shown in the paper by K. Ikeda, A. Pillay, A. Tsuboi
[229] that the dense linear order is interpretable in any almostω-
categorical theory with three countable models. P. Tanovic [451]
has shown that the Ehrenfeucht example or the Peretyat'kin ex-
ample is interpretable in any theory with three countable models
having an innite set of pairwise distinct constants. L. Mayer [299]
has described possible numbers of countable models ofo-minimal
theories (the class ofo-minimal theories includes the classical ex-
amples of Ehrenfeucht theories). R. Wencel showed that small
theories of Boolean orderedo-minimal structures areω-categorical
[474]. A. Alibek and B. S. Baizhanov [62] found examples of weakly
o
-minimal theories with arbitrary nite and countably many count-
able models. A. Alibek, B. S. Baizhanov, T. S. Zambarnaya [63]
proved that there continuum many countable models for an ordered
theory with a quasi-successor. D. Ilic, S. Moconja, P. Tanovic
[239] constructed an example of innite non-ω-categorical group
with an additional structure and having an Ehrenfeucht theory.
S. Lempp and T. Slaman [288] have shown that the property of
Ehrenfeuchtness is
1
Π
-complete. W. Calvert, V. S. Harizanov,
1
J. F. Knight, S. Miller [136] have described the complexity of in-
dex sets of classical Ehrenfeucht theories. Constructive models of
Ehrenfeucht theories are considered in the works by C. J. Ash and
T. S. Millar [73], G. A. Omarova [325], B. Khusainov, A. Nies and
R. A. Shore [254], A. N. Gavryushkin [177]{[182], S. S. Goncharov
[189, 190], E. B. Fokina, V. S. Harizanov, A. G. Melnikov [173], de-
cidable Ehrenfeucht theories are in the papers by M. Morley [313],
T. S. Millar [305, 306], R. Reed [364], V. S. Harizanov [193].
The
Lachlan Problem
, that is solved in the book, is known
more than thirty years. As a direction to the solution of this prob-
INTR
ODUCTION AND HISTORICAL SURVEY
15
lem, for various subclasses of the class of stable theories, the ab-
sence of theoriesTwith
1 < I (T, ω) < ω
is shown. This ab-
sence has been proved for the class of uncountably categorical the-
ories (J. T. Baldwin, A. H. Lachlan [89]), for superstable theories
(A. H. Lachlan [278], D. Lascar [283], S. Shelah [375], J. Sae [370],
A. Pillay [341]), for theories with nonprincipal superstable types
(T. G. Mustan [316]), for stable theories, in which
dcl(∅)
are mod-
els (A. Pillay [340]), for normal theories (A. Pillay [341]), for weakly
normal (1-based) theories (A. Pillay [344]), for theories admitting
nite codings (E. Hrushovski [222]), for unions of pseudo-
superstable theories (A. Tsuboi [461]), for theories without dense
forking chains (B. Herwig, J. Loveys, A. Pillay, P. Tanovic,
F. O. Wagner [390]; P. Tanovic [447]). A. Tsuboi [460] has proved
that any Ehrenfeucht theory, being a countable union ofω-cate-
gorical theories, is unstable. Then he generalized this theorem in
[461], where he proved that any theoryT, represented as a union
ofω-categorical theories
Tn,
n ∈ ω
, has a formula dening an non-
identical dense order on some denable set. A. A. Vikent'ev [58]
has shown a heredity of non-Ehrenfeuchtness for extensions of non-
Ehrenfeucht formula restrictions. P. Tanovic [448] has shown that
any stable theory, interpreting an innite set of pairwise distinct
constants, is non-Ehrenfeucht. He also has proved [450], that if
a theoryTis Ehrenfeucht then the set
dcl(∅)
is nite or the the-
oryThas the strict order property. A. Tsuboi [463] generalized
the result of P. Tanovic on innite
dcl(∅)
and proved that any
theory with three countable models and innite family of strongly
orthogonal formulas is unstable.
The development of Theory of Simple Theories (see F. O. Wag-
ner [59]; Z. Chatzidakis, A. Pillay [143]; B. Kim, A. Pillay [255];
B. Kim [30, 257]; M. Pourmahdian [358]; S. Shelah [376]), along-
side with the Lachlan problem for stable theories, has generated
the similar
Lachlan problem for simple theories
. B. Kim [257] has
generalized the Lachlan theorem (see A. H. Lachlan [278]) on su-
perstable theories and has shown that the class of supersimple
theories does not contain Ehrenfeucht theories.
So-called
powerful
types, that always are represented in Ehren-
feucht theories (see M. Benda [125]), play an important role for the
nding of number of countable models. In essence, the proof of the
16
INTR
ODUCTION AND HISTORICAL SURVEY
absence of Ehrenfeucht theories in aforesaid classes is reduced to
the assertion that these classes do not contain theories with non-
principal powerful types. Other essential properties, that Ehren-
feucht theories possess, are the non-symmetry of the semi-isolation
relation on nonempty sets of realizations of powerful types, and in-
nite weight of nonprincipal powerful types in simple theories (see
A. Pillay [341]; B. Kim [257]). The principles of systematization
for structural properties of Ehrenfeucht theories and their powerful
types have created in the candidate dissertation by the author [49].
A. H. Lachlan [279] introduced the notion of pseudo-plane and
has proved that structures of innite pseudo-planes are contained
in models ofω-categorical stable non-superstable theories. A. Pil-
lay [344] has obtained a similar result for stable non-1-based theo-
ries. Thus, the positive solution of the Lachlan problem is possible
only in the class of theories interpreting pseudo-planes.
Interrelations of types in theories are dened, in many aspects,
by the Rudin{Keisler preorders (see M. E. Rudin [366]). These pre-
orders have nitely many equivalence classes for Ehrenfeucht theo-
ries. D. Lascar [283]{[286] has investigated various Rudin{Keisler
preorders and has shown that any powerful type corresponds to a
greatest equivalence class with respect to Rudin{Keisler preorder.
E. Hrushovski [225], in 1988, using a modication of generic
Jonsson{Frasse construction
sson [245, 246]), has disproved the
(see R. Frasse [17, 174, 175]; B. Jon-
Zilber Conjecture
constructing
examples of strongly minimal not locally modular theories, in
which innite groups are not interpreted. His original construc-
tion, which served as a basis for building of appropriate examples
and solving other known model-theoretic problems, has given an
impetus to intensive studies of both the
Hrushovski construction
to-
gether with its various (in a broad sense) modications, capable of
creating \pathological" theories with given properties (see U. An-
drews [1, 65, 66, 67]; J. T. Baldwin [92]{[95], [97, 99, 106, 107];
J. T. Baldwin, A. S. Kolesnikov, S. Shelah [108]; J. T. Bald-
win, M. Koerwien, M. S. Laskowski [109]; Z. Ghadernezhad [21];
M. Hils [26]; S. V. Sudoplatov [54, 399, 408]; V. V. Verbovskiy
[57]; J. T. Baldwin, S. Shelah [98]; J. T. Baldwin, K. Holland [100,
101, 103, 105]; A. Baudisch [111, 112]; A. Baudisch, A. Martin-
Pizarro, M. Ziegler [116, 117, 118, 119, 121]; A. Baudisch, M. Hils,
INTR
ODUCTION AND HISTORICAL SURVEY
17
A. Martin-Pizarro, F. O. Wagner [120]; I. Ben Yaacov [124];
M. J. de Bonis, A. Nesin [130]; J. D. Caycedo, M. Hils [140];
J. D. Caycedo [141]; J. D. Caycedo, B. I. Zilber [142]; O. Cha-
puis, E. Hrushovski , P. Koiran, B. P. Poizat [282]; D. M. Evans
[164, 166]; D. M. Evans, M. E. Pantano [165]; D. M. Evans, M.
Wing Ho Wong[167]; D. M. Evans, Z. Ghadernezhad, K. Tent [170];
D. M. Evans, J. Hubicka, J. Nesetril [171, 172]; Z. Ghadernezhad,
K. Tent [183]; V. Guingona, C. D. Hill, L. Scow [192]; A. Hasson
[25, 197, 199, 201]; Z. Ghadernezhad [185]; A. Hasson, M. Hils
[200]; A. Hasson, E. Hrushovski [198]; A. Hasson, O. Mermel-
stein [202, 203]; B. Herwig [206]{[208]; B. Herwig, D. Lascar [209];
M. Hils [214]; K. Holland [217, 218]; E. Hrushovski [221, 223, 224];
E. Hrushovski, B. I. Zilber [226]; K. Ikeda [230, 231]; B. Kim,
A. S. Kolesnikov, A. Tsuboi [258]; A. S. Kolesnikov [31]; N. Peat-
eld, B. I. Zilber [334]; O. Mermelstein [301, 302]; G. Paolini
G.[332]; A. Pillay, A. Tsuboi [347]; B. P. Poizat [349], [350]; S. She-
lah [378]; K. Tent, M. Ziegler [456]; K. Tent [455, 457]; K. Tent,
B. I. Zilber [458]; V. V. Verbovskiy [466]; V. V. Verbovskiy, I. Yo-
neda [467]; A. Villaveces, P. Zambrano [470]; I. Yoneda [477];
M. Ziegler [478, 479]; B. I. Zilber [480]{[484]), and axiomatic bases
and their applications, allowing to determine applicability bounds
for that construction (see Yu. Anbo [64]; A. C. J. Bonato [6];
R. D. Aref'ev, J. T. Baldwin, M. Mazucco [72]; J. T. Baldwin [97]{
[104]; J. T. Baldwin, N. Shi [96]; A. Baudisch [114]; Z. Chatzidakis,
A. Pillay [143]; D. M. Evans [164]; D. M. Evans, M. S. Ferreira
[16, 168, 169]; J. B. Goode [191]; C. D. Hill [211, 212, 213]; K. Hol-
land [219]; K. Ikeda, A. Pillay, H. Kikyo A. Tsuboi, [232]; K. Ikeda
[235, 237]; K. Ikeda, H. Kikyo [234, 236, 238]; D. W. Kueker,
M. S. Laskowski [269]; M. S. Laskowski [287]; H. D. MacPherson,
K. Tent [293]; B. P. Poizat [351]; M. Pourmahdian [358]; R. Rajani
[363]; A. Valizadeh, M. Pourmahdian [464]; F. O. Wagner [471]{
[473]).
Relatively the Lachlan problem, B. Herwig [206] has shown
a fruitfulness of the Hrushovski construction realizing, using it,
a small stable theory with a type having innite weight. At the
same time, K. Ikeda [233] has shown that for an ordinary case, the
semantic approach for the Hrushovski construction cannot produce
a stable Ehrenfeucht theory.
18
INTR
ODUCTION AND HISTORICAL SURVEY
Intensive investigations of the original Jonsson{Frasse construc-
tion and its modications allowed to construct series of important
examples, to get structural and classication results: D. Bilge
[5]; G. Cherlin [9, 153]; E. Yu. Daniyarova, A. G. Myasnikov,
V. N. Remeslennikov [12]; A. Pongracz [44]; K. Schoretsanitis [46];
D. N. Siniora [48]; O. Angel, A. S. Kechris, R. Lyons [71]; A. Baud-
isch [113, 115, 122]; V. Bhat, V. Rodl [126]; V. Bhat, J. Nesetril,
C. Reiher, V. Rodl [127]; M. Bodirsky, M. Pinsker, T. Tsankov
[128]; M. Bodirsky [129]; P. J. Cameron [137]; E. Casanovas, R. Pe-
laez, M. Ziegler [138]; G. Cherlin, P. Komjath [144]; G. Cher-
lin, N. Shi [145, 146, 149]; G. Cherlin, N. Shi, L. Tallgren [147];
G. Cherlin, S. Shelah, N. Shi [148]; G. Cherlin, L. Tallgren [150];
G. Cherlin, S. Shelah [151, 152]; G. Conant, C. Terry [154]; G. Co-
nant [155]; G. Conant, A. Kruckman [156]; B. F. Csima, V. S. Ha-
rizanov, R. Miller, A. Montalban [157]; C. J. Eagle, I. Farah,
B. Hart, B. Kadets, V. Kalashnyk, M. Lupini [159]; C. W. Henson
[204, 205]; J. Hubicka, J. Nesetril [227, 228]; A. A. Ivanov [242, 243,
244]; Z. Kabluchko, K. Tent [247, 248]; I. Kaplan, P. Simon [249];
A. S. Kechris, V. G. Pestov, S. Todorcevic [250]; A. S. Kechris,
C. Rosendal [251]; A. S. Kechris, M. Sokic, S. Todorcevic [252];
A. Kruckman [261]; W. Kubis [262, 263, 264, 265]; A. H. Lach-
lan, R. Woodrow [280]; A. H. Lachlan [281]; H. D. MacPherson,
M. Pouzet, R. Woodrow [291]; H. D. MacPherson [294]; Sh. Ma-
sumoto [297, 298]; J. Melleray, T. Tsankov [300]; I. Muller [315];
J. Nesetril [317]; V. G. Puzarenko [362]; J. Schmerl [373]; K. Slut-
sky [383]; M. Sokic [384]; S. Solecki [385]; A. I. Stukachev [392];
A. M. Vershik [468, 469]; A. Zucker [485].
A topological approach to the classication of countable models
of a theory is proposed in the paper by G. Hjorth and A. S. Kechris
[215]. An inuence of non-elementary languages to the classica-
tion of structures and the independence property of axiomatizing
formulas in these languages, including Frasse-type constructions,
are investigated in the papers by G. Hjorth, I. A. Souldatos [216];
I. Rezniko [365]; I. A. Souldatos [386]{[389]. Z. Ghadernezhad,
H. Khalilian, M. Pourmahdian [184] investigated correspondences
between extreme amenability and amenability of automorphism
groups of Frasse{Hrushovski generic structures.
INTR
ODUCTION AND HISTORICAL SURVEY
19
Now we pass to the content of results in seven basic Chapters
of the book.
Chapter 1 begins (Section 1.1) with a syntactic characteriza-
tion of the class of complete theories with nitely many count-
able models on the basis of Rudin{Keisler preorders and distri-
bution functions of numbers of limit models. The basic part of
this characterization is distributed to the class of Ehrenfeucht the-
ories. Section 1.2 is devoted to the denitions of some basic cases
of inessential combinations and colorings of models, used below for
descriptions of intermediate constructions and for the solution of a
series of problems. We dene, in Section 1.3, the concept of type
reducibility, according to which a structure of type of predicate the-
ory is invariant with respect to restrictions of saturated structures
to the sets of realizations of the type. It is shown that the type
reducibility property does not hold in stable Ehrenfeucht theories.
An example, realizing the absence of the type reducibility in the
class of stable theories, has constructed. The notion of powerful
digraph is dened in Section 1.4. These digraphs, alongside with
powerful types, are always locally presented in Ehrenfeucht struc-
tures. Connections of powerful digraphs and powerful types (al-
ways presented in Ehrenfeucht structures) are shown, and proper-
ties of structures with powerful digraphs are investigated. We prove
in Section 1.5 the Tsuboi theorem [460, 461] on non-representation
of an Ehrenfeucht theory without denable nonidentical dense or-
ders and, in particular, without the strict order property by a union
of countably categorical theories. We prove a generalization of the
Tsuboi theorem [461] on non-representability of Ehrenfeucht the-
ory as a union of pseudo-superstable theories and of Kim theorem
[257] on the absence of Ehrenfeucht supersimple theories. This
generalization includes the Baldwin{Lachlan theorem on number
of countable models for
ω1-categorical theories [89], the Lachlan
theorem [278] on nonexistence of Ehrenfeucht theories in the class
of superstable theories, the analogous results proven by A. Pillay
[341, 344] for normal and1-based theories, and by E. Hrushovski
[222] for theories admitting nite codings.
The basic results, represented in Chapter 1, are published in
[393, 395, 396, 400, 401, 404, 460, 461]. Some modications of con-
20
INTR
ODUCTION AND HISTORICAL SURVEY
tent of Chapter 1 in [51], being reected in Chapter 1 of present
book, are taken from the paper [87] by B. S. Baizhanov, V. V. Ver-
bovskiy, and S. V. Sudoplatov. A helpful interpretation of results
is presented in the notes by E. Casanovas [139].
We describe (became already classical) semantic generic con-
structions in Chapter 2 (Section 2.1). Used for the solution of
the Lachlan problem, syntactic generic constructions, generaliz-
ing semantic constructions, are dened in Section 2.2. Properties
of various classes of syntactic generic constructions are considered
in Sections 2.3{2.5. Various kinds of fusions of generic construc-
tions are investigated, that also used for constructions of required
Ehrenfeucht theories (Section 2.6). In Sections 2.7 and 2.8 we show
some properties of generic algebras and principles for varieties of
generative classes. Basic results, described in Chapter 2, are stated
in [407, 411, 326, 402].
A series of Sections in Chapters 1 and 2 is written on the base
of Chapter 1 of the author's thesis for the doctor degree [50].
In Chapter 3, distributions of isolating and semi-isolating bi-
nary formulas linking realizations of types are investigated [327,
382, 420, 422, 423, 424]. In Section 3.1, we dene a class of alge-
bras distributing binary isolating formulas and introduce prelim-
inary denitions, notations, and properties of algebras connected
with relations of isolation and semi-isolation. In Sections 3.2, we
describe some basic examples for these algebras and for types bas-
ing these algebras. In Section 3.3, we dene a groupoid
P
ν(p)
of
principal formulas on the set of all realizations of a1-typep(as-
suming that there is an atomic model over a realization ofp) with
respect to a
alent principal formulas
|= p(a)
P
. In Section 3.4, we collect the basic properties of groupoids
and the signicant subgroupoids of
ν(p)
regular
labelling function
ϕ(a, y)
for which
ν(p)
for pairwise nonequiv-
ϕ(a, x) ` p(x)
P
. In Section 3.5,
ν(p)
holds,
using the successively-annihilating sums we construct two kinds of
monoids
P
containing an arbitrary group. In Section 3.6, we
ν(p)
produce a list of properties characterizing the class of groupoids
P
. Features of these groupoids for the class of special theories
ν(p)
are exposed in Section 3.7. In Section 3.8, we dene the notion
of join of groupoids and show the mechanism of extension of basic
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