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Серия «Математика» 2016. Т. 16. С. 131—144
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ИЗВЕСТИЯ
Иркутского государственного университета
УДК 510.67:515.12
MSC 03C30, 03C15, 03C50, 54A05
Closures and generating sets
related to combinations of structures *
S. V. Sudoplatov
Sobolev Institute of Mathematics, Novosibirsk State Technical University, Novosibirsk State University, Institute of Mathematics and Mathematical Modeling
Abstract. We investigate closure operators and describe their properties for E-combinations and P-combinations of structures and their theories including the negation of finite character and the exchange property. It is shown that closure operators for E-combinations correspond to the closures with respect to the ultraproduct operator forming Hausdorff topological spaces. It is also shown that closure operators for disjoint P-combinations form topological To-spaces, which can be not Hausdorff. Thus topologies for E-combinations and P-combinations are rather different. We prove, for E-combi-nations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set: it is shown that elements of the least generating set are isolated and dense in its E-closure.
Related properties for P-combinations are considered: it is proved that again the existence of a minimal generating set of theories is equivalent to the existence of the least generating set but it is not equivalent to the isolation of elements in the generating set. It is shown that P-closures with the least generating sets are connected with families which are not w-reconstructible, as well as with families having finite e-spectra.
Two questions on the least generating sets for E-combinations and P-combinations are formulated and partial answers are suggested.
Keywords: E-combination, P-combination, closure operator, generating set.
1. Introduction and preliminaries
Topological aspects related to model theoretic problems are investigated in a series of papers [1; 7; 8; 9; 10; 13; 14]. At present paper we study
* The research is partially supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (Grant NSh-6848.2016.1) and by Committee of Science in Education and Science Ministry of the Republic of Kazakhstan (Grant No. 0830/GF4).
structural properties of ^-combinations and P-combinations of structures and their theories [15] from the topological viewpoint.
In Section 2, using the E-operators and P-operators we introduce topologies (related to topologies in [1]) and investigate their properties.
In Section 3, we prove, for ^-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set. Related properties for P-combinations are considered.
Throughout the paper we use the following terminology in [15].
Definition 1. Let P = (Pi)ieI, be a family of nonempty unary predicates, (Ai)iei 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 iei
of the 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)ieI if Ai = (Ap \ Ai) \ S(Ai), i £ I. Structures A', which are elementary equivalent to CombP(Ai)i£I, will be also considered as P-combinations.
Clearly, all structures A = CombP(Ai)iei are represented as unions of
their restrictions Ai = (A' \ Pi) \ S(Ai) if and only if the set p^(x) =
{-iPi(x) | i £ 1} is inconsistent. If А' ф Comhp(Ai)iei, we write A! =
Combp (Ai) ie/u{oo}) where A^ = A! \ П Pi, maybe applying Morleyza-
iei
tion. Moreover, we write Combp(Ai)ieIu{^} for Combp(Ai)iei with the empty structure A^>.
Note that if all predicates Pi are disjoint, a structure Ap is a P-combi-nation and a disjoint union of structures Aj,. In this case the P-combination Ap is called disjoint. Clearly, for any disjoint P-combination Ap, Th(Ap) = Th(A'p), where A'p is obtained from Ap replacing Ai by pairwise disjoint Ai = Ai, i £ I. Thus, in this case, similar to structures the P-operator works for the theories Ti = Th(Ai) producing the theory Tp = Th(Ap), which is denoted by Combp(Ti)ieI.
Definition 2. 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)ieI to AE is the E-operator. The structure Ae is also called the E-combination of the structures Ai and denoted by CombE(Ai)ieI; here Ai = (Ae \ Ai) \ E(Ai), i £ I. Similar above, structures A', which are elementary equivalent to AE, are denoted by CombE(Aj)j, where Aj are restrictions of A to its E-classes.
Clearly, A = Ap realizing p^(x) is not elementary embeddable into Ap and can not be represented as a disjoint P-combination of Ai = Ai,
i G I .At the same time, there are ^-combinations such that all A = AE can be represented as E-combinations of some Aj = Ai. We call this representability of A to be the E-representability.
Definition 3. If there is A = Ae which is not E-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.
Definition 4. 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(AE). The value sup{e-Sp(A')) I A = Ae} is called the e-spectrum of the theory Th(AE) and denoted by e-Sp(Th(AE)).
Definition 5. If Ae does not have E-classes Ai, which can be removed, with all E-classes Aj = Ai, preserving the theory Th(AE), then AE is called e-prime, or e-minimal.
For a structure A = AE we denote by TH(A') the set of all theories Th(Ai) of E-classes Ai in A'.
By the definition, an e-minimal structure A consists of E-classes with a minimal set TH(A'). If TH(A') is the least for models of Th(A') then A is called e-least.
2. Closure operators
Definition 6. Let T be the class of all complete elementary theories of relational languages. For a set T C T we denote by C1E(T) the set of all theories Th(A), where A is a structure of some E-class in A = AE, Ae = CombE(A)iei, Th(A^) € T. As usual, if T = ClE(T) then T is said to be E-closed.
By the definition,
ClE (T) = TH(AE/), (2.1)
where AE, is an e-largest model of Th(AE), AE consists of E-classes representing models of all theories in T.
Note that the equality (2.1) does not depend on the choice of e-largest model of Th(Ae).
The following proposition is obvious.
Proposition 1. (1) If To, Ti are sets of theories, To Q T\ С T, then To Q CIe(To) с CIE(Ti).
(2) For any setT cT,T С C\E(T) if and only if the structure composed by E-classes of models of theories in T is not e-largest.
(3) Every finite set T cT is E-closed.
(4) (Negation of finite character) For any T £ ClE(T) \ T there are no finite T0 cT such that T £ ClE (T0).
(5) Any intersection of E-closed sets is E-closed.
For a set T С T of theories in a language E and for a sentence (p with Е(у) с E we denote by Tp the set {T £ T | у £T}. Denote by TF the family of all sets Tp.
Clearly, the partially ordered set (TF; Q) forms a Boolean algebra with the least element 0 = T-(x-x), the greatest element T = T(x~x), and operations Л, V, " satisfying the following equalities: Tp t\T^ = Tp^-ф),
Tp V Т-ф = T(pV^)i Tp = T—p■
By the definition, Tp QT- if and only if for any model M of a theory in T satisfying у we have M |= ф.
Proposition 2. IfT CT is an infinite set arnlT e T\T thenT e CI E(T) (i.e., T is an accumulation point for T with respect to E-closure ClE) if and only if for any formula у £T the set Tp is infinite.
Proof. Assume that there is a formula у £ T such that only finitely many theories in T, say T1,■■■, Tn, satisfy у. Since T / T then there is ф £T such that ф £T1 U --.и Tn. Then (у Л ф) £T does not belong to all theories in T. Since (у Л ф) does not satisfy E-classes in models of TE = CombE(Ti)ieI, where T = {Ti | i£ I}, we have T £ CIe(T).
If for any formula у £ T, Tp is infinite then {уЕ | у £ T}UTe (where уЕ are E-relativizations of the formulas у) is locally satisfied and so satisfied. Since TE is a complete theory then {уЕ | у £ T} c TE and hence T £ C1E(T). □
Proposition 2 shows that the closure ClE corresponds to the closure with respect to the ultraproduct operator [2; 3; 4; 6].
Theorem 1. For any sets %,T С T, C\E(ToUTi) = C\E(To)UC\E(Ti).
Proof. We have Cl e (To) U CIe (Ti) Q CIe (To UTi) by Proposition 1 (1).
Let T £ ClE(70 UT1) and we argue to show that T £ ClE(70) U ClE(T1). Without loss of generality we assume that T £ T0 UT1 and by Proposition 1, (3), T0 U T is infinite. Define a function f: T ^ P({0,1}) by the following rule: f (у) is the set of indexes k £ {0,1} such that у belongs
to infinitely many theories in Tk. Note that f (p) is always nonempty since by Proposition 2, p belong to infinitely many theories in T0 UT\_ and so to infinitely many theories in T0 or to infinitely many theories in T\. Again by Proposition 2 we have to prove that 0 € f (p) for each formula p € T or 1 € f (p) for each formula p € T. Assuming on contrary, there are formulas p,0 € T such that f (p) = {0} and f (0) = {1}. Since (p A0) € T and f (p A 0) is nonempty we have 0 € f (p A 0) or 1 € f (p A 0). In the first case, since TPAp Q Tp we get 0 € f (0). In the second case, since TPAp Q Tp we get 1 € f (p). Both cases contradict the assumption. Thus, TGC1e(TO)UC1e(TI). □
Corollary 1. (Exchange property) If Ti € CIe(TU {T2}) \ CIe(T) then T2 € CIe(TU{Ti}).
Proof. Since Ti € CIe (TU {T2}) = CIe (T) U {T2} by Proposition 1, (3) and Theorem 1, and <£ CIE(T), then = T2 and T2 e C1E(TU {Ti}) in view of Proposition 1, (1). □
Definition 7. [5]. A topological space is a pair (X, O) consisting of a set X and a family O of open subsets of X satisfying the following conditions:
(01) 0 €O and X € O;
(02) If Ui € O and U2 € O then Ui n U2 € O;
(03) If O' Q O then UO' € O.
Definition 8. [5]. A topological space (X, O) is a T0-space if for any pair of distinct elements xi,x2 € X there is an open set U € O containing exactly one of these elements.
Definition 9. [5]. A topological space (X, O) is Hausdorff if for any pair of distinct points xi,x2 € X there are open sets Ui,U2 € O such that xi € Ui, x2 € U2, and Ui n U2 = 0.
Let T C T be a set, Oe{T) = {T\C\E(T') | T' C T}. Proposition 1 and Theorem 1 imply that the axioms (01)-(03) are satisfied. Moreover, since for any theory TeT, CIE({T}) = {T} and hence, T\ClE({T}) = T{T} is an open set containing all theories in T, which are not equal to T, then (T, OE(T)) is a To-space. Moreover, it is Hausdorff. Indeed, taking two distinct theories Ti,T2 € T we have a formula p such that p € Ti and —p € T2. By Proposition 2 we have that Tp and T-p are closed containing Ti and T2 respectively; at the same time Tp and T-p form a partition of T, so Tp and T-p are disjoint open sets. Thus we have
Theorem 2. For any set T C T the pair (T, Oe(T)) is a Hausdorff topological space.
Similarly to the operator ClE(T) we define the operator Clp (T) for families P of predicates Pi as follows.
Definition 10. For a set T C T we denote by Clp(T) the set of all theories Th(A) such that Th(A) G T or A is a structure of type pM(x) in A = Ap, where Ap = Combp(Ai)iei and Th(A^) G T are pairwise distinct. As above, if T = ClP(T) then T is said to be P-closed.
Using above only disjoint P-combinations Ap we get the closure Clp(T) being a subset of Clp (T).
The following example illustrates the difference between Clp(T) and Clp (T).
Example 1. Taking disjoint copies of predicates Pi = {a G M0 | a < ci} with their <-structures as in [15, Example 4.8], Clp(T)\Tproduces models of the Ehrenfeucht example and unboundedly many connected components each of which is a copy of a model of the Ehrenfeucht example. At the same time Clp(T) produces two new structures: densely ordered structures with and without the least element.
The following proposition is obvious.
Proposition 3. (1) If To, Ti are sets of theories, % ç 71 C T, then To Q Clp (To) ç Clp (Ti ).
(2) Every finite set T cT is P-closed.
(3) (Negation of finite character) For any T G Clp (T) \ T there are no finite T0 cT such that T G Clp (70).
(4) Any intersection of P-closed sets is P-closed.
Remark 1. Note that an analogue of Proposition 2 for P-combinations fails. Indeed, taking disjoint predicates Pi, i G u, with 2i + 1 elements and with structures Ai of the empty language, we get, for the set T of theories Ti = Th(Ai), that Clp(T) consists of the theories whose models have cardinalities witnessing all ordinals in u+1. Thus, for instance, theories in T do not contain the formula
3x, y(-(x & y) A Vz((z & x) V (z & y))) (2.2)
whereas Clp(T) (which is equal to Clp(T)) contains a theory with the formula (2.2).
More generally, for Clp (T) with infinite T, we have the following.
Since there are no links between distinct Pi, the structures of pM(x) are defined as disjoint unions of connected components C(a), for a realizing p^ (x), where each C(a) consists of a set of realizations of pœ-preserving formulas 0(a,x) (i.e., of formulas p(a,x) with 0(a,x) h pœ(x)). Similar to Proposition 2 theories (a) of C(a)-restrictions of Aœ coincide and are characterized by the following property: T^^ia) G Clp(T) if and only if T^,c(a) G T or for any formula p G T^,C(a), there are infinitely many
theories T in T such that y satisfies all structures approximating C(a)-restrictions of models of T.
Thus similarly to Theorem 1, Corollary 1, and Theorem 2 we get the following three assertions for disjoint P-combinations.
Theorem 3. For any sets %,Ti C T, C\dP(T0 uTi) = CldP(%) U Clp(Ti).
Corollary 2. (Exchange property) If Ti G Clp(Tu {T2}) \ Clp(T) then T2 G Clp(Tu{Ti}).
Let T C T be a set, OdP(T) = {T\ Clp(T') \ V ç T}.
Theorem 4. For any set T C T the pair (T,Op(T)) is a topological T0 -space.
Remark 2. By Proposition 3, (2), for any finite T the spaces (T, Op(T)) and (T, Of,(T)) are Hausdorff, moreover, here OP(T) = Of,(T) consisting of all subsets of T. However, in general, the spaces (T, OP(T)) and (T, Op (T)) are not Hausdorff.
Indeed, consider structures Ai, i G I, where I = (w+1)\{0}, of the empty language and such that \Ai\ = i. Let Ti = Th(Ai), i G I, T = {Ti | i G I}. Coding the theories Ti by their indexes we have the following. For any finite set F c I, ClP(F) = Clp (F) = F, and for any infinite set INF C I, Clp (INF) = Clp (INF) = I. So any open set U is either cofinite or empty. Thus any two nonempty open sets are not disjoint.
Remark 3. If the closure operator Cldpr is obtained from Clp permitting repetitions of structures for predicates Pi, we can lose both the property of To-space and the identical closure for finite sets of theories. Indeed, for the example in Remark 2, Clpr(T) is equal to the Clp, -closure of any singleton {T} G Clpr (T) since the type p<x(x) has arbitrarily many realizations producing models for each element in T. Thus there are only two possibilities for open sets U: either U = 0 or U = T.
Remark 4. Let Tfin be the class of all theories for finite structures. By compactness, for a set T c Tfin, ClE(T) is a subset of Tfin if and only if models of T have bounded cardinalities, whereas Clp(T) is a subset of 7fln if and only if T is finite. Proposition 2 and its P-analogue allows to describe both Cle(T) and Clp(T), in particular, the sets Cle(T) \7fln and Clp(T) \ 7fln. Clearly, there is a broad class of theories in T which do not lay in
(J ClE(T) u U Clp (T).
T cTfln T cTfin
For instance, finitely axiomatizable theories with infinite models can not be approximated by theories in Tfin in such way.
Remark 5. Proposition 2 shows that if a set T has theories only with models in an axiomatizable class K then theories in CIe(T) again have models only in K. At the same time, by Remark 1, this assertion does not hold for P-closures.
3. Generating subsets of E-closed sets
Definition 11. Let T0 be a closed set in a topological space (T, OE(T)). A subset T' QT0 is said to be generating if T0 = ClE(T'). The generating set T' (for T0) is minimal if T' does not contain proper generating subsets. A minimal generating set T0' is least if T0' is contained in each generating set for T .
Remark 6. Each set T0 has a generating subset Ti with a cardinality < max{|£|,w}, where £ is the union of the languages for the theories in T0. Indeed, the theory T = Th(^.E), whose E-classes are models for theories in ClE(T0), has a model M with IM| < max{|£|,w}. The E-classes of M are models of theories in ClE(T0) and the set of these theories is the required generating set.
Theorem 5. If Tq is a generating set for a E-closed set T0 then the following conditions are equivalent:
(1) T' is the least generating set for T0;
(2) T' is a minimal generating set for T0;
(3) any theory in Ti is isolated by some set (Ti)p, i.e., for any T €TT there is p € T such that T')p = {T};
(4) any theory in Ti is isolated by some set (T0)p, i.e., for any T €TT there is p € T such that (T0)p = {T}.
Proof. (1) ^ (2) and (4) ^ (3) are obvious.
(2) ^ (1). Assume that Ti is minimal but not least. Then there is a generating set T' such that T¿\T'' = 0 and T'\T' = 0. Take T €T'\T''.
We assert that T € ClET \ {T}), i.e., T is an accumulation point of T'\{T}. Indeed, since T'\T0' = 0 and T' C ClET), then by Proposition 1, (3), Ti is infinite and by Proposition 2 it suffices to prove that for any p € T, T \ {T})p is infinite. Assume on contrary that for some p € T, (To \ {T})p is finite. Then (Ti)p is finite and, moreover, as Ti is generating for T0, by Proposition 2, (T0)p is finite, too. So (T^ )p is finite and, again by Proposition 2, T does not belong to ClE(T^ ) contradicting to CIe(TH) = T .
Since T € ClE(T' \ {T}) and TJ is generating for T0, then T' \ {T} is also generating for T contradicting the minimality of T0'.
(2) ^ (3). If Tl is finite then by Proposition 1, (3), T' = T0. Since T0 is finite then for any T €T0 there is a formula p € T negating all theories
in 70 \ {T}. Therefore, (70= (70% is a singleton containing T and thus, (70% isolates T.
Now let 70' be infinite. Assume that some T E T7 is not isolated by the sets (70%. It implies that for any y E T, (70' \ {T})p is infinite. Using Proposition 2 we obtain T E ClE(70' \ {T}) contradicting the minimality of 7 .
(3) ^ (2). Assume that
any theory T in 70' is isolated by some set (70%. By Proposition 2 it implies that T E ClE(70' \ {T}). Thus, 70' is a minimal generating set for 7 .
(3) ^ (4) is obvious for finite 70'. If 70' is infinite and any theory T in 70' is isolated by some set (70% then T is isolated by the set (70, since otherwise using Proposition 2 and the property that 7 ' generates 7 , there are infinitely many theories in 70' containing y contradicting |(70% | = 1. □
The equivalences (2) ^ (3) ^ (4) in Theorem 5 were noticed by E.A. Pa-lyutin.
Theorem 5 immediately implies
Corollary 3. For any structure AE, AE is e-minimal if and only if AE is e-least.
Definition 12. Let T be the theory Th(AE), where AE = CombE(Ai)ieI, {Th(Ai) | i E I} = 70. We say that T has a minimal/least generating set if CIe(70) has a minimal/least generating set.
Since by Theorem 5 the notions of minimality and to be least coincide in the context, below we shall consider least generating sets as well as e-least structures in cases of minimal generating sets.
Proposition 4. For any closed nonempty set 70 in a topological space (7, OE(7)) and for any 70' Q 70, the following conditions are equivalent:
(1) 70' is the least generating set for 70;
(2) any/some structure AE = CombE(Ai)i^i, where {Th(Ai) | i E I} = 70', is an e-least model of the theory Th(AE) and E-classes of each/some e-largest model of Th(AE) form models of all theories in 70;
(3) any/some structure AE = CombE(Ai)i^i, where {Th(Ai) | i E I} = 70', Ai ^ Aj for i = j, is an e-least model of the theory Th(AE), where E-classes of AE form models of the least set of theories and E-classes of each/some e-largest model of Th(AE) form models of all theories in 70.
Proof. (1) ^ (2). Let 70' be the least generating set for 70. Consider the structure Ae = CombE(Ai)iei, where {Th(Ai) | i E I} = 70'. Since 70' is the least generating set for 70, then AE is an e-least model of the theory Th(AE). Moreover, by Proposition 2, E-classes of models of Th(AE) form
models of all theories in To. Thus, E-classes of AE form models of the least set Tq of theories such that E-classes of each/some e-largest model of Th(AE) form models of all theories in To•
Similarly, constructing AE with Ai ф Aj for i = j, we obtain (1) ^ (3).
Since (3) is a particular case of (2), we have (2) ^ (3).
(3) ^ (1). Let Ae be an e-least model of the theory Th(AE) and E-classes of each/some e-largest model of Th(AE) form models of all theories in %■ Then by the definition of C1E, Tq is the least generating set for To- □
Note that any prime structure AE (or a structure with finitely many E-classes, or a prime structure extended by finitely many E-classes), is e-minimal forming, by its E-classes, the least generating set T' of theories for the set To of theories corresponding to E-classes of e-largest A'E ф Ae. Indeed, if a set T' is generating for T0 then by Proposition 2 there is a model M of T consisting of E-classes with the set of models such that their theories form the set T' ■ Since Ae prime (or with finitely many E-classes, or a prime structure extended by finitely many E-classes), then Ae is elementary embeddable into M (respectively, has E-classes with theories forming Tq , or elementary embeddable to a restriction without finitely many E-classes), then T' ç Tq , and so T' is the least generating set for T . Thus, Proposition 4 implies
Corollary 4. Any theory Th(AE) with a prime model M, or with a finite set {Th(Ai) I i G I}, or both with E-classes for M and Ai, has the least generating set.
Clearly, the converse for prime models does not hold, since finite sets T0 are least generating whereas theories in T can be arbitrary, in particular, without prime models. Again the converse for finite sets does not hold since there are prime models with infinite T0. Finally the general converse is not true since we can combine a theory T having a prime model with infinite To and a theory T' with infinitely many E-classes of disjoint languages and without prime models for these classes. Denoting by T' the set of theories for these E-classes, we get the least infinite generating set To UTO' for the combination of T and T', which does not have a prime model.
Replacing E-combinations by P-combinations we obtain the notions of (minimal/least) generating set for Clp(To).
Example in Remark 2 shows that Corollary 4 does not hold even for disjoint P-combinations. Indeed, take structures Ai, i G (ш + 1) \ {0}, in the remark and the theories Ti = Th(Ai) forming the Clp-closed set T. Since T is generated by any its infinite subset, we get that having prime models of Th(Ap), the closure Clp(T) does not have minimal generating sets.
For the example above, with the empty language, Cldpr (T) is generated by any singleton {T} G Clpr (T) since the type p^(x) has arbitrarily many
realizations producing models for each Ti, i E (u + 1) \ {0}. Thus, each element of Cldpr (7) forms a minimal generating set.
Natural questions arise concerning minimal generating sets:
Question 1. What are characterizations for the existence of least generating sets?
Question 2. Is there exists a theory Th(AE) (respectively Th(AP)) without the least generating set?
Remark 7. Obviously, for E-combinations, Question 1 has an answer in terms of Proposition 2 (clarified in Theorem 5) taking the least, under inclusion, set 70' generating the set ClE(70'). It means that 70' does not have accumulation points inside 70' (with respect to the sets (70%), i.e., any element in 70' is isolated by some formula, whereas each element T in CIe(70') \70' is an accumulation point of 70' (again with respect to (70%), i.e., 70' is dense in its E-closure.
A positive answer to Question 2 for ClP is obtained in Remark 2. Moreover, Theorem 5 does not hold with respect to the operator ClpP. Indeed, the theories Ti for the structures Ai, i E (u + 1) \ {0}, form the ClpP-closed set 70. Clearly, the theories Ti, for finite i, are isolated by formulas describing cardinalities for Ai, whereas 70 does not have minimal generating sets since it is generated by a subset 70' if and only if 70' is infinite.
More generally, if Ai consist of finitely many isomorphic definable equivalence classes and the number of these classes in unbounded varying the indexes i (taking, for instance, models of cubic theories [11; 12] with a fixed finite diameter, or isomorphic trees with a fixed finite diameter), then, as above, the P-closure 70 of the set of theories Th(Ai) does not have minimal generating sets.
Remark 7 shows that Theorem 5 fails for the operator Clp. At the same time using approximations of C(a)-restrictions of A^> in the arguments for (2) ^ (1) in Theorem 5 we get
Theorem 6. If 70' is a generating set for a P-closed set 70 with respect to the operator ClpP, then the following conditions are equivalent:
(1) 70' is the least generating set for 70,
(2) 70' is a minimal generating set for 70.
Definition 13. An infinite P-closed family 7 of theories is called (P,u)-reconstructible (respectively (P,d,u)-reconstructible) if 7 = ClP(70) (7 = Clp(70)) for any countable 70 Q7.
Since ClP(70) Q ClP(70) for any family 70, each (P, d, u)-reconstructible family is (P, u)-reconstructible.
By the definition, the families of theories in Remark 7 are (P, d, u)-reconstructible and therefore (P, u)-reconstructible.
Proposition 5. If a P-closed family T has a least generating set then T is not (P,w)-reconstructible.
Proof. It suffices to note that if T is (P, w)-reconstructible then T has only infinite generating sets T0 and for any T <eT0, T0\ {T}, being infinite, is generating for T as well. □
In contrast to Remark 7 we have
Proposition 6. If for a theory T = Th(AP ) e-Sp(T) is finite then the set T of theories for substructures Ai in A1 = Ap with respect to the predicates Pi and to the type p^(x) has a least generating set.
Proof. If e-Sp(T) is finite then for any generating set T for T we have \T\%\ < e-Sp(T). Thus, removing at most e-Sp(T) theories in T we get a minimal generating set for T being the least by Theorem 6. □
Propositions 5 and 6 imply
Corollary 5. If for a theory T = Th(AP), e-Sp(T) is finite then the set T of theories for substructures Ai in A1 = AP with respect to the predicates Pi and to the type pœ(x) is not (P,u)-rexonstructible.
References
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8. Kechris A. S. Fraisse limits, Ramsey theory, and topological dynamics of automorphism groups / A. S. Kechris, V. G. Pestov, S. Todorcevic // Geometric and Functional Analysis. — 2005. — Vol. 15, No. 1. — P. 106-189.
9. Newelski L. Topological dynamics of definable group actions / L. Newelski // J. Symbolic Logic. — 2009. — Vol. 74, No. 1. — P. 50-72.
10. Pillay A. Topological dynamics and definable groups / A. Pillay //J. Symbolic Logic. — 2013. — Vol. 78, No. 2. — P. 657-666.
11. Sudoplatov S. V. Group polygonometries / S. V. Sudoplatov. — Novosibirsk : NSTU, 2011, 2013. — 302 p. [in Russian]
12. Sudoplatov S. V. Models of cubic theories / S. V. Sudoplatov // Bulletin of the Section of Logic. — 2014. — Vol. 43, No. 1-2. — P. 19-34.
13. Sudoplatov S. V. Classes of structures and their generic limits / S. V. Sudoplatov // Lobachevskii J. Math. — 2015. — Vol. 36, No. 4. — P. 426-433.
14. Sudoplatov S. V. Semi-isolation and the strict order property / S. V. Sudoplatov, Tanovic P. // Notre Dame Journal of Formal Logic. — 2015. — Vol. 56, No. 4. — P. 555-572.
15. Sudoplatov S. V. Combinations of structures / S. V. Sudoplatov. — arXiv:1601.00041v1 [math.LO]. — 2016. — 19 p.
Sudoplatov Sergey Vladimirovich, Doctor of Sciences (Physics and Mathematics), docent; Leading researcher, Sobolev Institute of Mathematics SB RAS, 4, Academician Koptyug Avenue, Novosibirsk, 630090, tel.: (383)3297586; Head of Chair, Novosibirsk State Technical University, 20, K. Marx Avenue, Novosibirsk, 630073, tel.: (383)3461166; Docent, Novosibirsk State University, 1, Pirogova street, Novosibirsk, 630090, tel.: (383)3634020; Principal Researcher, Institute of Mathematics and Mathematical Modeling, 125, Pushkina Street, Almaty, Kazakhstan, 050010, tel.: +7(727)2720046 (e-mail: sudoplat@math.nsc.ru)
С. В. Судоплатов
Замыкания и порождающие множества, связанные с совмещениями систем
Аннотация. Исследуются операторы замыкания и описываются их свойства для В-совмещений и Р-совмещений систем и их теорий, включая отрицание конечного характера и свойство замены. Показано, что операторы замыкания для В-совмещений соответствуют замыканию относительно оператора ультрапроизведений и образуют хаусдорфовы топологические пространства. Также показано, что операторы замыкания для дизъюнктных Р-совмещений образуют топологические Го-пространства, которые могут не быть хаусдорфовыми. Таким образом, топологии для В-совмещений и Р-совмещений существенно различаются. Для В-совмещений доказано, что существование минимального порождающего множества теорий эквивалентно существованию наименьшего порождающего множества. Кроме того, синтаксически и семантически охарактеризовано свойство существования наименьшего порождающего множества: показано, что элементы наименьшего порождающего множества изолированы и являются плотными в своем В-замыкании.
Рассмотрены подобные свойства для Р-совмещений: доказано, что снова существование минимального порождающего множества теорий эквивалентно существованию наименьшего порождающего множества, но это не эквивалентно изолированности элементов в порождающем множестве. Показано, что Р-замыкания с наименьшими порождающими множествами связаны с семействами, которые не являются ш-восстановимыми, а также с семействами, имеющими конечный е-спектр.
Сформулированы два вопроса о наименьших порождающих множествах для В-совмещений и Р-совмещений. Предложены частичные ответы на эти вопросы.
Ключевые слова: E-совмещение, Р-совмещение, оператор замыкания, порождающее множество.
References
1. Baldwin J. T. A topology for the space of countable models of a first order theory / J. T. Baldwin, J. M. Plotkin // Zeitshrift Math. Logik and Grundlagen der Math. -1974. - Vol. 20, N 8-12. - P. 173-178.
2. Bankston P. Ulptraproducts in topology / P. Bankston // General Topology and its Applications. - 1977. - Vol. 7, N 3. - P. 283-308.
3. Bankston P. A survey of ultraproduct constructions in general topology / P. Bankston // Topology Atlas Invited Contributions. — 2003. — Vol. 8, N 2. -P. 1-32.
4. Кейслер Г. Теория моделей / Г. Кейслер, Ч. Ч. Чэн. - М. : Мир, 1977. - 616 с.
5. Энгелькинг Р. Общая топология / Р. Энгелькинг. - М. : Мир, 1986. - 752 с.
6. Ершов Ю. Л. Математическая логика / Ю. Л. Ершов, Е. А. Палютин. - М. : ФИЗМАТЛИТ, 2011. - 356 с.
7. Gismatullin J. On compactifications and the topological dynamics of definable groups / J. Gismatullin, D. Penazzi, A. Pillay // Annals of Pure and Applied Logic. - 2014. Vol. 165, N 2. - P. 552-562.
8. Kechris A. S. Fraïsse limits, Ramsey theory, and topological dynamics of automorphism groups / A. S. Kechris, V. G. Pestov, S. Todorcevic // Geometric and Functional Analysis. - 2005. - Vol. 15, N 1. - P. 106-189.
9. Newelski L. Topological dynamics of definable group actions / L. Newelski // J. Symbolic Logic. - 2009. - Vol. 74, N 1. - P. 50-72.
10. Pillay A. Topological dynamics and definable groups / A. Pillay // J. Symbolic Logic. - 2013. - Vol. 78, N 2. - P. 657-666.
11. Судоплатов С. В. Полигонометрии групп / С. В. Судоплатов. - Новосибирск : Изд-во НГТУ, 2011, 2013. - 302 с.
12. Sudoplatov S. V. Models of cubic theories / S. V. Sudoplatov // Bulletin of the Section of Logic. - 2014. - Vol. 43, N 1-2. - P. 19-34.
13. Sudoplatov S. V. Classes of structures and their generic limits / S. V. Sudoplatov // Lobachevskii J. Math. - 2015. - Vol. 36, N 4. - P. 426-433.
14. Sudoplatov S. V. Semi-isolation and the strict order property / S. V. Sudoplatov, P. Tanovic // Notre Dame Journal of Formal Logic. - 2015. - Vol. 56, N 4. -P. 555-572.
15. Sudoplatov S. V. Combinations of structures / S. V. Sudoplatov. -arXiv:1601.00041v1 [math.LO]. - 2016. - 19 p.
Судоплатов Сергей Владимирович, доктор физико-математических наук, доцент; ведущий научный сотрудник, Институт математики им. С. Л. Соболева СО РАН, 630090, Новосибирск, пр. Академика Коптюга, 4, тел.: (383)3297586; заведующий, кафедра алгебры и математической логики, Новосибирский государственный технический университет, 630073, Новосибирск, пр. К. Маркса, 20, тел.: (383)3461166; доцент, кафедра алгебры и математической логики, Новосибирский государственный университет, 630090, Новосибирск, ул. Пирогова, 1, тел.: (383)3634020; главный научный сотрудник, Институт математики и математического моделирования МОН РК, 050010, Казахстан, Алматы, ул. Пушкина, 125, тел.: +7(727)2720046 (e-mail: sudoplat@math.nsc.ru)
