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Серия «Математика»
2018. Т. 24. С. 82-101
УДК 510.67
MSG 03С30, 03С15, 03С50
DOI https://doi.org/10.26516/1997-7670.2018.24.82
Combinations of structures *
S. V. Sudoplatov
Sobolev Institute of Mathematics, Novosibirsk State Technical University, Novosibirsk State University, Novosibirsk, Russian Federation
Abstract. We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving w-cate-goricity and Ehrenfeuchtness under these combinations are characterized. The notions of e-spectra. are introduced and possibilities for e-spectra. are described.
It is shown that w-categoricity for disjoint P-combinations means that there are finitely many indexes for new unary predicates and each structure in new unary predicate is either finite or w-categorical. Similarly, the theory of ^-combination is w-categorical if and only if each given structure is either finite or w-categorical and the set of indexes is either finite, or it is infinite and Ei-classes do not approximate infinitely many n-types for n £ to. The theory of disjoint P-combination is Ehrenfeucht if and only if the set of indexes is finite, each given structure is either finite, or w-categorical, or Ehrenfeucht, and some given structure is Ehrenfeucht.
Variations of structures related to combinations and P-representability are considered. We introduce e-spectra for P-combinations and ^-combinations, and show that these e-spectra can have arbitrary cardinalities.
The property of Ehrenfeuchtness for ^-combinations is characterized in terms of e-spectra.
Keywords: combination of structures, P-combina.tion, P-combina.tion, e-spectrum.
The aim of the paper is to introduce operators (similar to [9; 10; 12; 14]) on classes of structures producing structures approximating given structure, as well as to study properties of these operators. These operators are
* The research is partially supported by Russian Foundation for Basic Researches (Grant No. 17-01-00531) and by Committee of Science in Education and Science Ministry of the Republic of Kazakhstan (Grant No. AP05132546).
1. Introduction
connected with natural topological properties related to families of theories [2-4; 7; 8].
In Section 2 we define P-operators, ^-operators, and corresponding combinations of structures. In Section 3 we characterize the preservation of w-categoricity for P-combinations and ^-combinations as well as Ehrenfeuchtness for P-combinations. In Section 4 we pose and investigate questions on variations of structures under P-operators and ^-operators. The notions of e-spectra for P-operators and ^-operators are introduced in Section 5. Here values for e-spectra are described. In Section 6 the preservation of Ehrenfeuchtness for ^-combinations is characterized.
Throughout the paper we consider structures of relational languages.
2. P-operators, ^-operators, combinations
Let P = (Pi)ie/, be a family of nonempty unary predicates, {Ai)i&i be
a family of structures such that Pi is the universe of Ai, i € I, and the
symbols Pi are disjoint with languages for the structures Aj, j € I. The
structure Ap ^ U Ai expanded by the predicates Pi is the P-union of the iei
structures Ai, and the operator mapping {Ai)i&i to Ap is the P-operator. The structure Ap is called the P-combination of the structures Ai and denoted by Combp{Ai)i&i if Ai = {Ap \ Ai) [ Yï{Ai), i € I. Structures A', which are elementary equivalent to Combp{Ai)i&i, will be also considered as P-combinations.
By the definition, without loss of generality we can assume for
Combp(A)ie/
that all languages T,{Ai) coincide interpreting new predicate symbols for
Ai by empty relation.
Clearly, all structures A! = Combp{Ai)i&i are represented as unions of
their restrictions Ali = {A' \ Pi) [ X {Ai) if and only if the set Poo{x) =
{-iPi{x) | i € 1} is inconsistent. If A! / Combp(^)ie/, we write A! =
Combp(^)ie/U{00}, where A!^ = A! \ H Pi, maybe applying Morleyza-
i&i
tion. Moreover, we write Combp(.4.i)ie/U{00} for Combp(^li)ie/ with the empty structure ^loo.
Notice that each structure A in a predicate language S can be represented as a P-combination. Indeed, taking formulas <fi{x), whose sets of solutions cover A, we can take (^¿-restrictions Ai of A with Pi{x) = <fi{x). The P-combination of Ai restricted to S forms A.
Clearly, if all predicates Pi are disjoint, a structure Ap is a P-combination and a disjoint union of structures Ai [14]. In this case the P-combination Ap is called disjoint. Clearly, for any disjoint P-combination Ap,
Th(^4p) = Th(^4p), where A'P is obtained from Ap replacing Ai by pairwise disjoint A'i = Ai, i € I. Thus, in this case, similar to structures the P-operator works for the theories T = Th(^) producing the theory Tp = Th(^lp), which is denoted by Combp{T)i&j.
On the opposite side, if all Pi coincide then Pi{x) = (x & x) and removing the symbols Pi we get the restriction of Ap which is the combination of the structures Ai [10; 12].
For an equivalence relation E replacing disjoint predicates Pi by E-classes we get the structure Ae being the E-union of the structures Ai. In this case the operator mapping {Ai)i^i to Ae is the E-operator. The structure Ae is also called the E-combination of the structures Ai and denoted by Combe^)^/; here Ai = {Ae \ Ai) \ T<{Ai), i € I. Similar above, structures A', which are elementary equivalent to Ae, are denoted by CombE{A!j)j&j, where Aj are restrictions of A! to its I?-classes.
If Ae < A!, the restriction A! \ {A1 \ Ae) is denoted by A!^. Clearly, A! = A!e\[A!00, where A'E = Comb E{A'i)i&i, Ali is a restriction of A! to its i?-class containing the universe Ai, i € I.
Considering an ^-combination i® we will identify I?-classes Ai with structures Ai.
Clearly, the nonempty structure A!^ exists if and only if I is infinite.
Notice that any _E-operator can be interpreted as P-operator replacing or naming _E-classes for Ai by unary predicates Pi. For infinite I, the difference between 'replacing' and 'naming' implies that „Aqo can have unique or unboundedly many _E-classes returning to the I?-operator.
Thus, for any ^-combination Ae, Th(„4.e) = TYv{A!E), where A'E is obtained from Ae replacing Ai by pairwise disjoint Ali = Ai, i € I. In this case, similar to structures the I?-operator works for the theories Ti = Th(^4j) producing the theory Te = Th(„4.e), which is denoted by Combe{Ti)i&j, by Te, or by CombeT, where T = {Ti \ i € I}.
Note that P-combinations and _E-unions can be interpreted by randomizations [1] of structures.
Sometimes we admit that combinations Combp(„4.j)je/ and Combe{Ai)izi are expanded by new relations or old relations are extended by new tuples. In these cases the combinations will be denoted by ECombp(^li)ie/ and ECombE{Ai)i&i, respectively
3. w-categoricity and Ehrenfeuchtness for combinations
Proposition 3.1. If predicates Pi are pairwise disjoint, the languages T:{Ai) are at most countable, i € I, |/| < w, and the structure Ap is infinite then the theory Th(„4p) is w-categorical if and only if I is finite and each structure Ai is either finite or w-categorical.
Proof. If I is infinite or there is an infinite structure Ai which is not w-categorical then T = Th(„4p) has infinitely many n-types, where n = 1 if | > oj and n = no for Th(^) with infinitely many no-types. Hence by Ryll-Nardzewski Theorem Th(„4p) is not w-categorical.
If Th(^4.p) is w-categorical then by Ryll-Nardzewski Theorem having finitely many n-types for each n € uj, we have both finitely many predicates Pi and finitely many n-types for each Pj-restriction, i. e., for Th(„4j). □
Notice that Proposition 3.1 is not true if a P-combination is not disjoint: taking, for instance, a graph A\ with a set P\ of vertices and with infinitely many Ei-edges such that all vertices have degree 1, as well as taking a graph A2 with the same set P\ of vertices and with infinitely many R2-edges such that all vertices have degree 1, we can choose edges such that R\ n R2 = 0, each vertex in Pi has (R\ U i?2)-degree 2, and alternating R\- and E2-edges there is an infinite sequence of (R\ U -fi^-edges. Thus, A\ and A2 are w-categorical whereas Comb(„4.i, A2) is not.
Note also that Proposition 3.1 does not hold replacing Ap by Ae• Indeed, taking infinitely many infinite I?-classes with structures of the empty languages we get an w-categorical structure of the equivalence relation E. At the same time, Proposition 3.1 is preserved if there are finitely many i?-classes. In general case Ae does not preserve the w-categoricity if and only if ^¿-classes approximate infinitely many n-types for some n € w, i. e., there are infinitely many n-types qm(x), rri € uj, such that for any m € u, <fj(x) € qj(x), j < m, and classes E^,..., Ekm, all formulas <fj(x) have
m
realizations in Ae\ U Ekr- Indeed, assuming that all Ai are w-categorical
r= 1
we can lose the w-categoricity for Th(^e) only having infinitely many retypes (for some n) inside ^loo. Since all n-types in ^loo are locally (for any formulas in these types) realized in infinitely many Ai, Ei-classes approximate infinitely many n-types and Th(^le) is not w-categorical. Thus, we have the following
Proposition 3.2. If the languages T:(Ai) are at most countable, i € I, |/| < oj, and the structure Ae is infinite then the theory Th(^le) is coca,tegorica,I if and only if each structure Ai is either finite or uj-categorical, and I is either finite, or infinite and Ei-classes do not approximate infinitely many n-types for any n € uj.
As usual we denote by I(T, A) the number of pairwise non-isomorphic models of T having the cardinality A.
Recall that a theory T is Ehrenfeucht if T has finitely many countable models (I(T,uj) < uj) but is not w-categorical (I(T,uj) > 1). A structure with an Ehrenfeucht theory is also Ehrenfeucht.
Theorem 3.3. If predicates Pi are pairwise disjoint, the languages T:(Ai) are at most countable, i € I, and the structure Ap is infinite then
the theory Th(.4.p) is Ehrenfeucht if and only if the following conditions hold:
(a) I is finite;
(b) each structure Ai is either finite, or w-categorical, or Ehrenfeucht;
(c) some Ai is Ehrenfeucht.
Proof. If I is finite, each structure Ai is either finite, or w-categorical, or Ehrenfeucht, and some Ai is Ehrenfeucht then T = Th(„4p) is Ehrenfeucht since each model of T is composed of disjoint models with universes Pi and
7(2» = J]/(Th(A),min{|^|,w}). (3.1)
Ш
Now if I is finite and all Ai are w-categorical then by (3.1), I(T, w) = 1, and if some I(Th(Ai),w) > w then again by (3.1), I(T, w) > oj.
Assuming that [/[ > w we have to show that the non-w-categorical theory T has infinitely many countable models. Assuming on contrary that I(T,w) < w, i. е., T is Ehrenfeucht, we have a nonisolated powerful type q(x) € S(T) [5], i. е., a type such that any model of T realizing q(x) realizes all types in S(T). By the construction of disjoint union, q(x) should have a realization of the type Poo(x) = {->Pi{x) \ i € I}. Moreover, if some Th(^) is not w-categorical for infinite Ai then q{x) should contain a powerful type of Th(^) and the restriction r(y) of q{x) to the coordinates realized by Poo(x) should be powerful for the theory Th(^oo), where Доо is infinite and saturated, as well as realizing r(y) in a model Л4 |= T, all types with coordinates satisfying Poo(x) should be realized in Л4 too. As shown in [11; 12], the type r(y) has the local realizability property and satisfies the following conditions: for each formula ip(y) € r(y), there exists a formula ip(y,z) of T (where l{y) = l(z)), satisfying the following conditions:
(i) for each a € r(M), the formula ip(a, y) is equivalent to a disjunction of principal formulas ipi(a,y), i < m, such that ipi(a,y) b r(y), and |= ipi(a,b) implies, that b does not semi-isolate a;
(ii) for every a, b € r(M), there exists a tuple с such that |= <p(c) Л ■0(c, а) Л ip(c, b).
Since the type Poo(x) is not isolated each formula <p(y) € r(y) has realizations J in (J Ai. On the other hand, as we consider the disjoint union of
ш
Ai and there are no non-trivial links between distinct Pi and Py, the sets of solutions for ip(d,y) with |= ip(d) in {->Pi{x) [[= Pi(dj) for some dj € d} are either equal or empty being composed by definable sets without parameters. If these sets are nonempty the item (i) can not be satisfied: ip(a,y) is not equivalent to a disjunction of principal formulas. Otherwise all -0-links for realizations of r(y) are situated inside the set of solutions for poo(y) = U Poo{yj)- In this case for a \= r{y) the formula 3z(tp(z, а) Л tp(z, y)) does
Vj&V
not cover the set r(M) since it does not cover each (^-approximation of r(M). Thus, the property (ii) fails.
Hence, (i) and (ii) can not be satisfied, there are no powerful types, and the theory T is not Ehrenfeucht. □
4. Variations of structures related to combinations and
I?-representability
Clearly, for a disjoint P-combination Ap with infinite I, there is a structure A! = Ap with a structure A!^. Since the type Poo(x) is nonisolated (omitted in Ap), the cardinalities for A!^ are unbounded. Infinite structures A!^ are not necessary elementary equivalent and can be both elementary equivalent to some Ai or not. For instance, if infinitely many structures Ai contain unary predicates Qo, say singletons, without unary predicates Qi and infinitely many Ay for %' / i contain Qi, say again singletons, without Qo then A!^ can contain Qo without Qi, Qi without Qo, or both Qo and QFor the latter case, A!^ is not elementary equivalent neither Ai, nor Ay.
A natural question arises:
Question 1. What can be the number of pairwise elementary non-equivalent structures A'^7
Considering an ^-combination Ae with infinite I, and all structures A! = Ae, there are two possibilities: each non-empty E-restriction of A!^, i. e. a restriction to some E-class, is elementary equivalent to some Ai, i € I, or some ^-restriction of A!^ is not elementary equivalent to all structures Ai, i € I.
Similarly Question 1 we have:
Question 2. What can be the number of pairwise elementary non-equivalent E-restrictions of structures A
Example 4.1. Let Ap be a disjoint P-combination with infinite I and composed by infinite Ai, i € I, such that I is a disjoint union of infinite Ij, j € J, where Aij contains only unary predicates and unique nonempty unary predicate Qj being a singleton. Then A!^ can contain any singleton Qj and finitely or infinitely many elements in P| Qj. Thus, there
jeJ
are 2lJl • (A + 1) non-isomorphic A'^, where A is a least upper bound for
cardinalities P| QJ
j&J
For T = Th(^lp), we denote by /^(T, A) the number of pairwise non-isomorphic structures A'qq having the cardinality A.
Clearly, /оо(Т, Л) < 1{Т, Л).
If structures А'oo exist and do not have links with A'P (for instance, for a disjoint P-combination) then Ioo(T, A) + l < I(T, A), since if models of T are isomorphic then their restrictions to Poo(x) are isomorphic too, and Poo(x) can be omitted producing A!^ = 0. Here /^(T, A) + 1 = I{T, A) if and
only if all /(Th(^),A) = 1 and, moreover, for any I IJ Pi )-restrictions
Kiel J
Bp,B'p of B,B' |= T respectively, where \B\ = \B'\ = A, and their P»-restrictions Bi, B[, there are isomorphisms /¿: Bi^B[ preserving Pi and with an isomorphism |J /¿: ВРЦ-В'р.
ш
The following example illustrates the equality /^(T, A) + 1 = I(T, A) with some /(Th(A),A) > l.
Example 4.2. Let Po be a unary predicate containing a copy of the Ehrenfeucht example [13] with a dense linear order < and an increasing chain of singletons coding constants Ck, к € w; Pn, n > 1, be pairwise disjoint unary predicates disjoint to Po such that P\ = (—oo,Cq) Pn+2 = [dn,c'n+1), new, and IJ Pn forms a universe of prime model (over 0) for
n> 1
another copy of the Ehrenfeucht example with a dense linear order <' and an increasing chain of constants c'fc, к € w. Now we extend the language
^ = (^j ^ ) P'rii {Ста}) {cri})n-€w
by a bijection / between Po = {a | a < Co or Co < a} and {a' | a' <' Cq or Cq <' a'} such that a < b f(a) <' f(b). The structures A!^ consist of realizations Poo(x) which are bijective with realizations of the type {cn < x | n € w}.
For the theory T of the described structure ECombp(A)ie/ we have /(T, uj) = 3 (as for the Ehrenfeucht example and the restriction of T to Po) and /oo(T,w) = 2 (witnessed by countable structures with least realizations of Poo(x) and by countable structure with realizations of Poo(x) all of which are not least).
For Example 4.1 of a theory T with singletons Qj in Ai and for a cardinality A > 1, we have
if J and A are finite;
if J is infinite and | J\ > A; if J is infinite and | J\ < A.
Clearly, A! = Ap realizing Poo(x) is not elementary embeddable into Ap and can not be represented as a disjoint P-combination of A[ = Ai, i € I. At the same time, there are ^-combinations such that all A! = Ae
min{| J|,A}
, £ cw
ЫТ, А) = < - ^^
2и,
can be represented as ^-combinations of some Aj = Ai. We call this representability of A! to be the E-representability. If, for instance, all Ai are infinite structures of the empty language then any A! = Ae is an ^-combination of some infinite structures A'j of the empty language too. Thus we have:
Question 3. What is a characterization of E-representability for all M = Ae'-
Definition (cf. [6]). For a first-order formula <p(xi,... ,xn), an equivalence relation E and a formula a(x) we define a (E, a)-relativized formula (pE'a by induction:
n
(i) if cp is an atomic formula then ¡pE'a = <p(xi,..., xn) A /\ E(xi, xf) A
3y(E(Xl,y) Aa(y));
(ii) if tp = tprx, where r € {A, V, —>}, and ipE'a and xE'a are defined then <pE'a = ipE'aTXE'a]
(iii) if <p(xi,... ,xn) = -iip(xi,... ,xn) and tpE'a(xi,... ,xn) is defined
n
then ipE>a{xi,...,xn) = ^ipE'a(xi,...,xn) A A (E(xi, Xj) A 3y(E(x\,y) A
¿,.7 = 1
(iv) if <p(xi,..., xn) = 3xtp(x, x\,..., xn) and tpE'a(x, x\,..., xn) is defined then
tpE'a(x\, ...,xn) = = 3x ^/\(E(x, Xi) A 3y(E(x, y) A a(y)) A ipE'a(x, xh..., xn)"j ;
(v) if <p(xi,... ,xn) = Vxip(x,xi,... ,xn) and ipE'a(x,xi,... ,xn) is defined then
(pE'a(xi, ...,xn) = = ~ix E(x, Xi) A 3y(E(x, y) A a(y)) ->■ ipE'a(x, xh..., xn)"j .
We write E instead of (E, a) if a = (x & x).
Note that two I?-classes Ei and Ej with structures Ai and Aj (of a language E), respectively, are not elementary equivalent if and only if there is a E-sentence tp such that Ae \ Ei \= (pE (with Ai \= <p) and Ae \ Ej |= (-■(p)E (with Aj |= -i<p). In this case, the formula tp is called (i, j)-separating.
The following properties are obvious:
(1) If tp is (i, j)-separating then ip is (j, i)-separating.
(2) If ¡p is (i, j)-separating and ip is (i, fc)-separating then ip A ip is both (i, j)-separating and (i, fc)-separating.
(3) There is a set of (i, j)-separating sentences, for j in some J ç I\{i}, which separates Ai from all structures Aj ^ Ai.
The set is called e-separating (for Ai) and Ai is e-separable (witnessed
by
Assuming that some A! = Ae is not I?-representable, we get an IS'-class with a structure B in A! which is e-separable from all Ai, i € I, by a set It means that for some sentences <pi with Ae \ Eî |= <ff, i- e., Ai \= <fi, the
( Y
sentences /\ <pi , where Io Çfln /, form a consistent set, satisfying
\i€lo J
the restriction of A! to the class E'B with the universe B of B. Thus, answering Question 3 we have
Proposition 4.3. For any E-combination Ae the following conditions are equivalent:
(1) there is A' = Ae which is not E-representable;
(2) there are sentences <pi such that Ai \= <fi, i € I, and the set of
( Y
sentences f\ -><pi , where Io Çfln I, is consistent with Th(^le).
\ieio J
Proposition 4.3 implies
Corollary 4.4. If Ae has only finitely many pairwise elementary non-equivalent E-classes then each A! = Ae is E-representable.
5. e-spectra
If there is A! = Ae which is not I?-representable, we have the E'~ representability replacing E by E' such that E' is obtained from E adding equivalence classes with models for all theories T, where T is a theory of a restriction B of a structure A! = Ae to some E-class and B is not elementary equivalent to the structures Ai. The resulting structure Ae> (with the -E'-representability) is a e-completion, or a e-saturation, of Ae-The structure Ae> itself is called e-complete, or e-saturated, or e-universal, or e-largest.
For a structure Ae the number of new structures with respect to the structures Ai, i. e., of the structures B which are pairwise elementary non-equivalent and elementary non-equivalent to the structures Ai, is called the e-spectrum of Ae and denoted by e-Sp(^e). The value sup{e-Sp(A)) | A! = Ae} is called the e-spectrum of the theory T1i(„4e0 and denoted by e-Sp(Th(^E)).
If Ae does not have I?-classes Ai, which can be removed, with all Isolasses Aj = Ai, preserving the theory T1i(„4e0, then Ae is called e-prime, or e-minimal.
For a structure A! = Ae we denote by TH(^l/) the set of all theories Th(A) of ^-classes Ai in A'.
By the definition, an e-minimal structure A! consists of I?-classes with a minimal set TH(„4')- If TH(„4') is the least for models of Th(„4') then A! is called e-least.
The following proposition is obvious:
Proposition 5.1. 1. For a given language 0 < e-Sp(Th(„4.e)) <
2max{|S|,w}
2. A structure Ae is e-largest if and only if e-Sp(AE) = 0. In particular, an e-minimal structure Ae is e-largest is and only ¿/e-Sp(Th(„4.e)) = 0.
3. Any weakly saturated structure Ae, i- e., a structure realizing all types o/Th(^e) is e-largest.
4. For any E-combination Ae, if A < e-Sp(Th(„4.e)) then there is a structure A! = Ae with e-Sp(^l/) = A; in particular, any theory Th(„4.e) has an e-largest model.
5. For any structure Ae, e-Sp(„4e) = |TH(A'E,) \ TH(„4e)|, where A'E, is an e-largest model o/Th(^e).
6. Any prime structure Ae is e-minimal (but not vice versa as the e-minimality is preserved, for instance, extending an infinite E-class of given structure to a greater cardinality). Any small theory Th(„4.e) has an e-minimal model (being prime), and in this case, the structure Ae is e-minimal if and only if
TK(Ae)= H THCA'),
A'=Ae
i. e., Ae is e-least.
7. If Ae is e-least then e-Sp(^e) = e-Sp(Th(„4.e)).
8. If e-Sp(Th(„4.e)) finite and Th(„4.e) has e-least model then Ae is e-minimal if and only if Ae is e-least and if and only if e-Sp(^e) = e-
Sp(Th(AE)).
9. If e-Sp(Th(„4.e)) is infinite then there are A! = Ae such that e-Sp(„4/) = e-Sp(Th(„4.e)) but A! is not e-minimal.
10. A countable e-minimal structure Ae is prime if and only if each E-class Ai is a prime structure.
Reformulating Proposition 3.2 we have
Proposition 5.2. For E-combinations which are not EComb, a countable theory Th(^le) without finite models is u-categorical if and only if e-Sp(Th(„4.e)) = 0 and each E-class Ai is either finite or uj-categorical.
Note that if there are no links between iS-classes (i. е., the Comb is considered, not EComb) and there is A! = Ae which is not iS-representable, then by Compactness the e-completion can vary adding arbitrary (finitely or infinitely) many new -Б-classes with a fixed structure which is not elementary equivalent to structures in old iS-classes.
Proposition 5.3. For any cardinality X there is a theory T = Th(^g) of a language E such that |E| = |A + 1| and e-Sp(T) = A.
Proof. Clearly, for structures Ai of fixed cardinality and with empty language we have e-Sp(Th(„4.e)) = 0. For A > 0 we take a language E consisting of unary predicate symbols Pi, i < A. Let Ai,n+1 be a structure having a universe Ayn with n elements and Pi = Ai,n, Pj = 0, i,j < A, i ф j, n € oj \ {0}. Clearly, the structure Ae, formed by all Ai,n, is e-minimal. It produces structures A! = Ae containing iS-classes with infinite predicates Pi, and structures of these classes are not elementary equivalent to the structures Ai,n- Thus, for the theory T = Th(^g) we have e-Sp(T) = A. □
In Proposition 5.3, we have e-Sp(T) = |E(T)|. At the same time the following proposition holds.
Proposition 5.4. For any infinite cardinality A there is a theory T = Th(^g) of a language E such that |E| = A and e-Sp(T) = 2Л.
Proof. Let Pj be unary predicate symbols, j < A, forming the language E, and Ai be structures consisting of only finitely many nonempty predicates Pj1,..., Pjk and such that these predicates are independent. Taking for the structures Ai all possibilities for cardinalities of sets of solutions for formulas P^1 (ж) Л... Л PJfcJfc (x), 5jt € {0,1}, we get an e-minimal structure Ae such that for the theory T = Th(„4e) we have e-Sp(T) = 2Л.
Another approach for e-Sp(T) = 2Л was suggested by E.A. Palyutin. Taking infinitely many Ai with arbitrarily finitely many disjoint singletons Rj1,..., Rjk, where E consists of Rj, j < A, we get AI = Ae with arbitrarily many singletons for any subset of A producing 2Л iS-classes which are pairwise elementary non-equivalent. □
If e-Sp(T) = 0 the theory T is called e-non-abnormalized or (e, 0)-abnormalized. Otherwise, i. е., if e-Sp(T) > 0, T is e-abnormalized. An e-abnormalized theory T with e-Sp(T) = A is called (e, X)-abnormalized. In particular, an (e, l)-abnormalized theory is e-categorical, an (e, ^-abnormalized theory with n € oj\ {0,1} is e-Ehrenfeucht, an (e, w)-abnormalized theory is e-countable, and an (e, 2A)-abnormalized theory is (e, X)-maximal.
If e-Sp(T) = A and T has a model Ae with e-Sp(^e) = t1 then Ae is called (e, к)-abnormalized, where к is the least cardinality with ¡jl + ж = A.
By proofs of Propositions 5.3 and 5.4 we have
Corollary 5.5. For any cardinalities ¡i < X and the least cardinality x with ¡jl + k = A there is an (e, X)-abnormalized theory T with an (e, x)-abnormalized model Ae-
Let Ae and Be' be structures and Ce>> = Ae\[Be' be their disjoint union, where E" = E]JE'. We denote by ComLim(„4.e, Be') the number of elementary pairwise non-equivalent structures V which are both a restriction of A! = Ae to some E-class and a restriction of B' = Be' to some -E'-class as well as V is not elementary equivalent to the structures Ai and Bj.
We have:
ComUm(AE,BE>) < min{e-Sp(Th(^E)), e-Sp(Th(£E/))},
max{e-Sp(Th(^e)), e-Sp(Th(£>e/))} < e-Sp(Th(CE//)), e-Sp(Th(„4.e)) + e-Sp(Th(£>e')) = e-Sp(Th((V)) + ComLim(^e, BE>).
Indeed, all structures witnessing the value e-Sp(Th(Ce»)) can be obtained by Th(^le) or Th{BE') and common structures are counted for
ComLim^e, BE')-
If Ae = Be' then ComLim(„4.e, £>£/) = e-Sp(Th(„4.e)). Assuming that Ae and Be' do not have elementary equivalent classes Ai and Bj, the number ComLim(„4.e, Be') can vary from 0 to 2lsl+w.
Indeed, if Th(^le) or Th(BE>) does not produce new, elementary non-equivalent classes then ComLim^e, BE') = 0. Otherwise we can take structures Ai and Bi with one unary predicate symbol P such that P has 2i elements for Ai and 2i + 1 elements for Bi, i G uj. In this case we have Sp(Th(^E)) = 1, Sp(Th(BE')) = 1, ComUm(AE,BE') = 1, and CE" witnessed by structures with infinite interpretations for P. Extending the language by unary predicates Pi, i < X, and interpreting Pi in disjoint structures as for P above, we get Sp(Th(^e)) = A, Sp(Th(£>e')) = A, ComLim(„4.E, BE') = A. Thus we have
Proposition 5.6. For any cardinality X there are structures Ae and Be' of a language E such that |E| = |A + 1| and ComLim(„4.e, BE') = A.
Applying proof of Proposition 5.4 with even and odd cardinalities for intersections of predicates in Ai and Bj respectively, we have Sp(Th(^e)) = 2\ Sp(Th(£>e')) = 2\ ComL\m(AE,BE<) = 2A. In particular, we get
Proposition 5.7. For any infinite cardinality X are structures Ae and Be' of a language E such that |E| = A and ComLim^e, BE') = 2A.
Replacing _E-classes by unary predicates Pi (not necessary disjoint) being universes for structures Ai and restricting models of Th(^4p) to the set of realizations of Poo(x) we get the e-spectrum e-Sp(Th(^4p)), i. e., the
number of pairwise elementary non-equivalent restrictions of M |= Th(„4p) to Poo(x). We also get the notions of (e, A)-abnormalized theory ТЬ(Лр), of (e, A)-abnormalized model of ТЬ(Лр), and related notions.
Note that for any countable theory T = Th(AP), e-Sp(T) < I(T,uj). In particular, if I(T, w) is finite then e-Sp(T) is finite too. Moreover, if T is w-categorical then e-Sp(T) = 0, and if T is an Ehrenfeucht theory, then e-Sp(T) < I(T,w). Illustrating the finiteness for Ehrenfeucht theories we consider
Example 5.8. Similar to Example 4.2, let To be the Ehrenfeucht theory of a structure Mo, formed from the structure (Q; <) by adding singletons Rk for elements Ck, Ck < Ck+1, к € и, such that lim Ck = oo. It is well
k—>oo
known that the theory T has exactly 3 pairwise non-isomorphic models:
(a) a prime model Mo ( lim Ck = 00);
k—>oo
(b) a prime model M\ over a realization of powerful type Poo(x) € S1(0), isolated by sets of formulas {ck < x | к € w};
(c) a saturated model M2 (the limit lim Ck is irrational).
k—>oo
Now we introduce unary predicates Pi = {a € Mo | a < Ci}, i < w, on Л40. The structures Ai = Mo I Pi form the P-combination Ap with the universe Mo. Realizations of the type Poo(x) in M1 and in M2 form two elementary non-equivalent structures Ac and A!^ respectively, where Доо has a dense linear order with a least element and A!^ has a dense linear order without endpoints. Thus, e-Sp(To) = 2 and To is e-Ehrenfeucht.
As E.A. Palyutin noticed, varying unary predicates Pi in the following way: P2i = {a e M0 \ a < c2ij, P2i+1 = {a € M0 | a < c2i+1}, we get е-8р(Тз) = 4 since the structures A!^ have dense linear orders with(out) least elements and with(out) greatest elements.
Modifying Example above, let Tn be the Ehrenfeucht theory of a structure Mn, formed from the structure (Q; <) by adding constants Ck, Ck < Cfc+i, к € и, such that lim Ck = 00, and unary predicates Ro, ■ ■ ■, Rn-2
k—>oo
which form a partition of the set Q of rationale, with
|= Vx, у ((x < y) 3z ((x < z) Л (z < у) Л Ri(z))), i = 0,..., n - 2.
The theory Tn has exactly n + 1 pairwise non-isomorphic models:
(a) a prime model Mn ( lim Ck = 00);
k—>oo
(b) prime models Mf over realizations of powerful types Pi{x) € S1(0), isolated by sets of formulas {ck < x | к € w} U {Pi(x)}, i = 0,..., n — 2 ( lim ck € Pi);
k—>oo
(c) a saturated model Minftyn (the limit lim Ck is irrational).
k—>oo
Now we introduce unary predicates Pi = {a € Mn | a < Ci}, i < w, on Mn. The structures Ai = Mn \ Pi form the P-combination Ap with the universe Mn. Realizations of the type Poo(x) in Mf and in M^ form n — 1
elementary non-equivalent structures A™, j <n — 2, and A1^, where A™ has a dense linear order with a least element in Rj, and A^ has a dense linear order without endpoints. Thus, e-Sp(Tra) = n and Tn is e-Ehrenfeucht.
Note that in the example above the type Poo{x) has n — 1 completions by formulas Ro(x),..., Rn-2{x).
Example 5.9. Taking a disjoint union Ai of m € oj \ {0} copies of Mo in the language {<j, Rk}j<m,k&uj and unary predicates Pi = {a \ A4 |= 3x(a < x A Ri{x))} we get the P-combination Ap with the universe M for the structures Ai = M \ Pi, i € oj. We have e-Sp(Th(^P)) = 3m - 1 since each connected component of Ai produces at most two possibilities for dense linear orders or can be empty on the set of realizations of poo(x), and at least one connected component has realizations of poo(x).
Marking the relations <j by the same symbol < we get the theory T with
Examples 5.8 and 5.9 illustrate that having a powerful type Poo{x) we get e-Sp(Th(^4p)) / 1, i. e., there are no e-categorical theories Th(^lp) with a powerful type Poo{x). Moreover, we have
Theorem 5.10. For any theory Th(^lp) with non-symmetric or definable semi-isolation on the complete type Poo{x), e-Sp(Th(„4p)) / 1.
Proof. Assuming the hypothesis we take a realization a of Poo(x) and construct step-by-step a (a,p00(x))-thrifty model J\f of Th(^lp), i. e., a model satisfying the following condition: if <p(x,y) is a formula such that cp(a, y) is consistent and there are no consistent formulas ip(a, y) with ip(a,y) h Poo(x) then ¡p{a,N) = 0.
At the same time, since Poo(x) is non-isolated, for any realization a of Poo(x) the set Poo{x) U {-><p(a,x) | <p(a,x) h is consistent. Then
there is a model TV' |= Th(^lp) realizing Poo(x) and which is not (a',poo(x))-thrifty for any realization a' of Poo{x).
If semi-isolation is non-symmetric, TV \ Poo(x) and TV' \ Poo{x) are not elementary equivalent since the formula ip(a, y) witnessing the non-symmetry of semi-isolation has solutions in TV' \ Poo(x) and does not have solutions in TV Ï Poo(x).
If semi-isolation is definable and witnessed by a formula ip(a,y) then again TV Ï Poo(x) and TV' \ Poo{x) are not elementary equivalent since -iip(a,y) is realized in TV' \ Poo{x) and it does not have solutions in TV \
Poo
Thus, e-Sp(Th(^P)) > 1. □
Since non-definable semi-isolation implies that there are infinitely many 2-types, we have
2=1
Corollary 5.11. For any theory Th(AP) with e-Sp(Th(„4P)) = 1 the structures А'ж are not w-categorical.
Applying modifications of the Ehrenfeucht example as well as constructions in [12], the results for e-spectra of ^-combinations are modified for P-combinations:
Proposition 5.12. For any cardinality X there is a theory T = Th(^p) of a language E such that |E| = max{A, w} and e-Sp(T) = A.
Proof. Clearly, if Poo(x) is inconsistent then e-Sp(T) = 0. Thus, the assertion holds for A = 0.
If A = 1 we take a theory T\ with disjoint unary predicates Pi, i € w, and a symmetric irreflexive binary relation R such that each vertex has -R-degree 2, each Pi has infinitely many connected components, and each connected component on Pi has diameter i. Now structures on Poo(x) have connected components of infinite diameter, all these structures are elementary equivalent, and e-Sp(Ti) = 1.
If A = n > 1 is finite, we take the theory Tn in Example 5.8 with e-Sp(Tra) = n, as well as we can take a generic Ehrenfeucht theory T'x with RK(T{) = 2 and with A — 1 limit model Mi over the type Poo(x), i < A — 1, such that each Mi has a Qj-chains, j < and does not have Qfc-chains for к > i. Restricting the limit models to Poo(x) we get A elementary non-equivalent structures including the prime structure TV0 without Qi-chains and structures Mi [ Poo(x), i < A — 1, which are elementary non-equivalent by distinct (non)existence of Qj-chains.
Similarly, taking A > uj disjoint binary predicates Rj for the Ehrenfeucht example in 5.8 we have A structures with least elements in Rj which are not elementary equivalent each other. Producing the theory T\ we have e-Sp(TA) = A.
Modifying the generic Ehrenfeucht example taking A binary predicates Qj with Qj-chains which do not imply Qfc-chains for к > i we get A elementary non-equivalent restrictions top00(x). □
Note that as in Example 5.8 the type Poo{x) for the Ehrenfeucht-like example T\ has A completions by the formulas Rj(x) whereas the type Poo{x) for the generic Ehrenfeucht theory is complete. At the same time having A completions for the p^(x)-restrictions related to T\, the Poo(x)-restrictions the generic Ehrenfeucht examples with complete Poo{x) can violet the uniqueness of the complete 1-type like the Ehrenfeucht example To, where Ac realizes two complete 1-types: the type of the least element and the type of elements which are not least.
Proposition 5.13. For any infinite cardinality X there is a theory T = Th(^p) of a language E such that |E| = A and e-Sp(T) = 2Л.
Proof. Let T be the theory of independent unary predicates Rj, j < X, (defined by the set of axioms Эх (R/C1(x) A ... A Rkm{x) A Ri1(x) A ... A
-iRin(x)), where {k\,..., km}n{l\,... ,ln} = 0) such that countably many of them form predicates Pi, i < uj, and infinitely many of them are independent with Pi. Thus, T can be considered as Th(^lp). Restrictions of models of T to sets of realizations of the type Poo(x) witness that predicates Rj distinct with all Pi are independent. Denote indexes of these predicates Rj by J. Since Poo(x) is non-isolated, for any family A = (5j)j&j, where
5j € {0,1}, the types q^(x) = {R^ | j € J} can be pairwise independently realized and omitted in structures M \ Poo(x) for M \= T. Then any predicate Rj can be independently realized and omitted in these restrictions. Thus there are 2A restrictions with distinct theories, i. e., e- Sp(T) = 2A. □
Since for ^-combinations Ae and P-combinations Ap and their limit structures Ao, being respectively structures on iS-classes and Poo(x), the theories Th(^l00) are defined by types restricted to E(x,y) and Poo(x), and for any countable theory there are either countably many types or continuum many types, Propositions 5.3, 5.4, 5.12, and 5.13 implies the following
Theorem 5.14. If T = Th(AE) (respectively, T = Th(AP)) is a countable theory then e-Sp(T) € oj U {w,2w}. All values in uj U {w,2w} have realizations in the class of countable theories of E-combinations (of P-combinations).
6. Ehrenfeuchtness for ^-combinations
Theorem 6.1. If the language (J is at most countable and the
i&i
structure Ae is infinite then the theory T = Th(^le) is Ehrenfeucht if and only if e-Sp(T) < oj (which is equivalent here to e-Sp(T) = 0) and for an e-largest model Ae> |= T consisting of E'-classes Aj, j € J, the following conditions hold:
(a) for any j € J, I(Th(Aj),uj) < uj;
(b) there are positively and finitely many j € J such that I(Th(Aj), uj) >
1/
(c) if I(Th(Aj), uj) < 1 then there are always finitely many Aj' = Aj or always infinitely many Aj' = Aj independent of Ae> |= T.
Proof. If e-Sp(T) < oj and the conditions (a)-(c) hold then the theory T is Ehrenfeucht since each countable model Ae" |= T is composed of disjoint models with universes E'l = A^, k € K, and I(T,uj) is a
e-Sp(T)
sum °f finitely many possibilities for models with I representa-
1=0
tives with respect to the elementary equivalence of ^"-classes that are
not presented in a prime (i. e., e-minimal) model of T. These possibilities are composed by finitely many possibilities of I(Th(Ak),uj) > 1 for Aw = Ak and finitely many of Aw ^ Ak with I(Th(Ak"), w) > 1. Moreover, there are C(I(Th(Ak),oj),mi) possibilities for substructures consisting of Ay = Ak where rrii is the number of I?-classes having the theory Th(^4fc), C(n, m) = C™+m_l is the number of combinations with repetitions for n-element sets with m places. The formula for I(T,uj) is based on the property that each i^'-class with the structure Ak can be replaced, preserving the elementary equivalence of AE", by arbitrary B = Ak-
Now we assume that the theory T is Ehrenfeucht. Since models of T with distinct theories of I?-classes are not isomorphic, we have e-Sp(T) < uj. Applying the formula for I(T,uj) we have the conditions (a), (b). The condition (c) holds since varying unboundedly many we get
I(T,UJ) > w.
The conditions e-Sp(T) < uj and e-Sp(T) = 0 are equivalent. Indeed, if e-Sp(T) > 0 then taking an e-minimal model M we get, by Compactness, unboundedly many _E-classes, which are elementary non-equivalent to E-classes in M. It implies that I(T,uj) > uj. □
Since any prime structure is e-minimal (but not vice versa as the e-minimality is preserved, for instance, extending an infinite I?-class of given structure to a greater cardinality preserving the elementary equivalence) and any Ehrenfeucht theory T, being small, has a prime model, any Ehrenfeucht theory Th(^e) has an e-minimal model.
We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving w-categoricity and Ehrenfeuchtness under these combinations are characterized. The notions of e-spectra are introduced and possibilities for e-spectra are described.
7. Conclusion
We introduced and studied combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving w-categoricity and Ehrenfeuchtness under these combinations are characterized. The notions of e-spectra are introduced and possibilities for e-spectra are described.
References
1. Andrews U., Keisler H.J. Separable models of randomizations. J. Symbolic Logic, 2015, vol. 80, no. 4, pp. 1149-1181.https://doi.org/10.1017/jsl.2015.33
2. Baldwin J.T., Plotkin J.M. A topology for the space of countable models of a first order theory. Zeitshrift Math. Logik and Grundlagen der Math., 1974, vol. 20, no. 8-12, pp. 173-178.https://doi.org/10.1002/malq.19740200806
3. Bankston P. Ulptraproducts in topology. General Topology and its Applications, 1977, vol. 7, no. 3, pp. 283-308.
4. Bankston P. A survey of ultraproduct constructions in general topology. Topology Atlas Invited Contributions, 2003, vol. 8, no. 2, pp. 1-32.
5. Benda M. Remarks on countable models. Fund. Math., 1974, vol. 81, no. 2, pp. 107-119.https: //doi.org/10.4064/fm-81-2-107-119
6. Henkin L. Relativization with respect to formulas and its use in proofs of independence. Composito Mathematica, 1968, vol. 20, pp. 88-106.
7. Newelski L. Topological dynamics of definable group actions. J. Symbolic Logic, 2009, vol. 74, no. 1, pp. 50-72.https://doi.org/10.2178/jsl/1231082302
8. Pillay A. Topological dynamics and definable groups. J. Symbolic Logic, 2013, vol. 78, no. 2, pp. 657-666.https://doi.org/10.2178/jsl.7802170
9. Sudoplatov S.V. Transitive arrangements of algebraic systems. Siberian Math. J., 1999, vol. 40, no. 6, pp. 1142-1145. https://doi.org/10.1007/BF02677538
10. Sudoplatov S.V. Inessential combinations and colorings of models. Siberian Math. J., 2003, vol. 44, no. 5, pp. 883-890.https://doi.org/10.1023/A:1025901223496
11. Sudoplatov S.V. Powerful digraphs. Siberian Math. J., 2007, vol. 48, no. 1, pp. 165-171.https://doi.org/10.1007/s11202-007-0017-1
12. Sudoplatov S.V. Klassifikatsiya schetnykh modeley polnykh teoriy [Classification of Countable Models of Complete Theories]. Novosibirsk, NSTU Publ., 2018.(in Russian)
13. Vaught R. Denumerable models of complete theories. Infinistic Methods, London, Pergamon, 1961, pp. 303-321.
14. Woodrow R.E. Theories with a finite number of countable models and a small language. Ph. D. Thesis. Simon Fraser University, 1976, 99 p.
Sudoplatov Sergey Vladimirovich, Doctor of Sciences (Physics and Mathematics), Associate Professor, Leading Researcher, Sobolev Institute of Mathematics SB RAS, 4, Academician Koptyug Avenue, Novosibirsk, 630090, Russian Federation, tel.: (383)3297586; Head of Chair, Novosibirsk State Technical University, 20, K. Marx Avenue, Novosibirsk, 630073, Russian Federation, tel.: (383)3461166; Professor, Novosibirsk State University, 1, Pirogov st., Novosibirsk, 630090, Russian Federation, tel.: (383)3634020 (e-mail: sudoplat@math.nsc.ru)
Received 19.04.2018
Комбинации структур
С. В. Судоплатов
Институт математики им. С. Л. Соболева СО РАН, Новосибирск, Российская Федерация
Новосибирский государственный технический университет, Новосибирск, Российская Федерация,
Новосибирский государственный университет, Новосибирск, Российская Федерация
Аннотация. Исследуются комбинации структур, для данных семейств структур, относительно семейств одноместных предикатов и отношений эквивалентности. Охарактеризованы условия сохранения ш-категоричности и эренфойхтовости для этих комбинаций. Введены понятия е-спектров и описаны возможности для е-спектров.
Показано, что ш-категоричность для дизъюнктных Р-комбинаций равносильна конечному числу индексов для новых одноместных предикатов с условием конечности или ш-категоричности каждой структуры в новых одноместных предикатах. Аналогично, теория Р-комбинации ш-категорична тогда и только тогда, когда каждая данная структура либо конечна, либо ш-категорична, и множество индексов либо конечно, либо бесконечно и при этом Pi-классы не аппроксимируют бесконечное число те-типов для п £ со. Теория дизъюнктной Р-комбинации эренфойхтова тогда и только тогда, когда множество индексов конечно, каждая данная структура либо конечна, либо ш-категорична, либо эренфойхтова, и некоторая структура эренфойхтова.
Рассмотрены вариации структур, относящиеся к комбинациям и Р-представи-мости.
Введены е-спектры для Р-комбинаций и Р-комбинаций, и показано, что эти е-спектры могут иметь произвольные мощности.
В терминах е-спектров охарактеризовано свойство эренфойхтовости для Р-ком-бинаций.
Ключевые слова: комбинация структур, Р-комбинация, е-спектр, Р-комбина-ция.
Список литературы
1. Andrews U. Separable models of randomizations // J. Symbolic Logic. 2015. Vol. 80, N 4. P. 1149-1181. https://doi.org/10.1017/jsl.2015.33
2. Baldwin J. Т., Plotkin J. M. A topology for the space of countable models of a first order theory // Zeitshrift Math. Logik and Grundlagen der Math. 1974. Vol. 20, N 8-12. P. 173-178. https://doi.org/10.1002/malq.19740200806
3. Bankston P. Ulptraproducts in topology // General Topology and its Applications. 1977. Vol. 7, N 3. P. 283-308.
4. Bankston P. A survey of ultraproduct constructions in general topology // Topology Atlas Invited Contributions. 2003. Vol. 8, N 2. P. 1-32.
5. Benda M. Remarks on countable models // Fund. Math. 1974. Vol. 81, N 2. P. 107119. https://doi.org/10.4064/fm-81-2-107-119
6. Henkin L. Relativization with respect to formulas and its use in proofs of independence // Composito Mathematica. 1968. Vol. 20. P. 88-106.
7. Newelski L. Topological dynamics of definable group actions //J. Symbolic Logic. 2009. Vol. 74, N 1. P. 50-72. https://doi.org/10.2178/jsl/1231082302
8. Pillay A. Topological dynamics and definable groups //J. Symbolic Logic. 2013. Vol. 78, N 2. P. 657-666. https://doi.org/10.2178/jsl.7802170
9. Судоплатов С. В. Транзитивные размещения алгебраических систем // Сиб. мат. журн. 1999. Т. 40, № 6. С. 1347-1351. https://doi.org/10.1007/BF02677538
10. Судоплатов С. В. Несущественные совмещения и раскраски моделей // Сиб. мат. журн. 2003. Т. 44, № 5. С. 1132-1141. https://doi.org/10.1023/ А: 1025901223496
11. Судоплатов С. В. Властные орграфы // Сиб. мат. журн. 2007. Т. 48, № 1. С. 205-213. https://doi.org/10.1007/sll202-007-0017-l
12. Судоплатов С. В. Классификация счетных моделей полных теорий. Новосибирск : НГТУ, 2018.
13. Vaught R. Denumerable models of complete theories // Infinistic Methods. London : Pergamon, 1961. P. 303-321.
14. Woodrow R. E. Theories with a finite number of countable models and a small language. Ph. D. Thesis. Simon Fraser University, 1976. 99 p.
Судоплатов Сергей Владимирович, доктор физико-математических наук, доцент; ведущий научный сотрудник, Институт математики им. С. JI. Соболева СО РАН, Российская Федерация,630090, Новосибирск, пр. Академика Коптюга, 4, тел.: (383)3297586; заведующий кафедрой алгебры и математической логики, Новосибирский государственный технический университет, Российская Федерация, 630073, Новосибирск, пр. К. Маркса, 20, тел. (383)3461166; профессор кафедры алгебры и математической логики, Новосибирский государственный университет, Российская Федерация, 630090, Новосибирск, ул. Пирогова, 1, тел. (383)3634020 (e-mail: sudoplat@math.nsc.ru)
Поступила в редакцию 19.04-2018
