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50
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
2019. Т. 28. С. 95-112
УДК 510.67:512.541
MSG 03С30, 03С15, 03С50, 54А05
DOI https://doi.org/10.26516/1997-7670.2019.28.95
Ranks for Families of Theories of Abelian Groups *
In. I. Pavlyuk
Novosibirsk State Pedagogical University, Novosibirsk, Russian Federation
S. V. Sudoplatov
Sobolev Institute of Mathematics, Novosibirsk State Technical University, Novosibirsk State University, Novosibirsk, Russian Federation
Abstract. The rank for families of theories is similar to Morley rank and can be considered as a measure for complexity or richness of these families. Increasing the rank by extensions of families we produce more rich families and obtaining families with the infinite rank that can be considered as "rich enough". In the paper, we realize ranks for families of theories of abelian groups. In particular, we study ranks and closures for families of theories of finite abelian groups observing that the set of theories of finite abelian groups in not totally transcendental, i.e., its rank equals infinity. We characterize pseudofinite abelian groups in terms of Szmielew invariants. Besides we characterize e-minimal families of theories of abelian groups both in terms of dimension, i.e., the number of independent limits for Szmielew invariants, and in terms of inequalities for Szmielew invariants. These characterizations are obtained both for finite abelian groups and in general case. Furthermore we give characterizations for approximability of theories of abelian groups and show the possibility to count Szmielew invariants via these parameters for approximations. We describe possibilities to form d-definable families of theories of abelian groups having given countable rank and degree.
Keywords: family of theories, abelian group, rank, degree, closure.
* This research was partially supported by the program of fundamental scientific researches of the SB RAS No. 1.1.1, project No. 0314-2019-0002 (Sections 1-4), Committee of Science in Education and Science Ministry of the Republic of Kazakhstan, Grant No. AP05132546 (Section 5), and Russian Foundation for Basic Researches, Project No. 17-01-00531-a (Section 6).
In. I. PAVLYUK, S. V. SUDOPLATOV 1. Introduction
The rank [11] for families of theories, similar to Morley rank, can be considered as a measure for complexity or richness of these families. Thus increasing the rank by extensions of families we produce more rich families and obtaining families with the infinite rank that can be considered as "rich enough". A series of such rich families, containing all families of given language, is described in [5]. Additional properties of families of theories defined by sets of sentences are studied in [6].
Links and closures for families of theories of abelian groups, as well as values of their e-spectra are described in [8].
In the present paper, we consider and realize ranks for families of theories of abelian groups. The paper is organized as follows. Preliminary notions, notations and related results for families of theories and for theories of abelian groups, including Szmielew invariants, are collected in Sections 2 and 3. In Section 4, we study closures and ranks for theories of finite abelian groups. In particular, we observe that the set of theories of finite abelian groups in not totally transcendental (Theorem 4.1) and characterize pseudofinite abelian groups in terms of Szmielew invariants (Theorem 4.2). In Section 5, we characterize e-minimal families of theories of abelian groups both in terms of dimension, i.e., the number of independent limits for Szmielew invariants, and in terms of inequalities for Szmielew invariants. These characterizations are obtained both for finite abelian groups (Theorems 5.1, 5.3, 5.4) and in general case (Theorem 5.5). Theorem 5.6 gives characterizations for approximability of theories of abelian groups, as well as it produce the possibility to count Szmielew invariants via these parameters for approximations. In Section 6, we describe possibilities to form d-definable families of theories of abelian groups having given countable rank and degree (Theorems 6.2 and 6.4).
2. Preliminaries
Throughout we consider families T of complete first-order theories of a language E = E(T). For a sentence (p we denote by Tip the set {T eT | <p € T} being the (p-neighbourhood in T.
Definition [12]. Let T be a family of theories and T be a theory, T фТ. The theory T is called T-approximated, or approximated by T, or T-approximable, or a pseudo-T-theory, if for any formula (p € T there is V € T such that ip eT'.
If T is T-approximated then T is called an approximating family for T, theories T'eT are approximations for T, and T is an accumulation point for T.
An approximating family T is called e-minimal if for any sentence ip € E(T), %p is finite or T-^ip is finite.
It was shown in [12] that any e-minimal family T has unique accumulation point T with respect to neighbourhoods Tp, and Tu {T} is also called e-minimal.
Following [11] we define the rankKS(-) for the families of theories, similar to Morley rank [7], and a hierarchy with respect to these ranks in the following way.
For the empty family T we put the rank RS(T) = —1, for finite nonempty families T we put RS(T) = 0, and for infinite families T — RS(T) > 1.
For a family T and an ordinal a = /3 + 1 we put RS(T) > a if there are pairwise inconsistent E(T)-sentences <pn, n € oj, such that RS(7^n) > /3, new.
If a is a limit ordinal then RS(T) > a if RS(T) > /3 for any /3 < a.
We set RS(T) = a if RS(T) > a and RS(T) £ a + 1.
If RS(T) > a for any a, we put RS(T) = oo.
A family T is called e-totally transcendental, or totally transcendental, if RS(T) is an ordinal.
Proposition 2.1 [11]. If an infinite family T does not have e-minimal subfamilies Tip then T is not totally transcendental.
If T is totally transcendental, with RS(T) = a > 0, we define the degree ds(T) of T as the maximal number of pairwise inconsistent sentences <pi such that RS(7^,i) = a.
Theorem 2.2 [11]. For any family T, RS(T) = RS(C1 E{T)), and ifT is nonempty and e-totally transcendental then ds(T) = ds(Cle(T)).
Theorem 2.3 [11]. For any family T with [E(T)[ < oj the following conditions are equivalent:
(1) |C1E(T)[=2";
(2) e-Sp(T) = 2";
(3) RS(T) = oo.
Definition [6]. Let T be a family of first-order complete theories in a language E. For a set $ of E-sentences we put 7$ = {TeT|T|=$}. A family of the form T<s> is called d-definable (in T). If $ is a singleton {<£>} then Tp = T<s> is called s-definable.
Theorem 2.4 [6]. LetT be a family of a countable language E and with RS(T) = oo, a € {0,1}, n € oj \ {0}. Then there is a d-definable subfamily T<s> such that RS(7$) = a and ds(7$) = n.
Recall that a subfamily % of T is called doo-definable if To is a union, possibly infinite, of d-definable subfamilies of T.
Theorem 2.5 [6]. Let T be a family of a countable language E and with RS(T) = oo, a be a countable ordinal, n € w\ {0}. Then there is a doc-definable subfamily T* С T such that RS(T*) = a and ds(T*) = n.
Similarly [7] and following [11], for a nonempty family T, we denote by B(T) the Boolean algebra consisting of all subfamilies where (p are sentences in the language E (T).
Theorem 2.6 [7; 11]. A nonempty family T is e-totally transcendental if and only if the Boolean algebra B(T) is superatomic.
Recall the definition of the Cantor-Bendixson rank. It is defined on the elements of a topological space X by induction: CBx(p) > 0 for all pel; CBx(p) > a if and only if for any /3 < a, p is an accumulation point of the points of CB^-rank at least /3. We set CBx(p) = a if and only if both CBx(p) > a and CBx(p) jt a + 1 hold; if such an ordinal a does not exist then CBx(p) = oo. Isolated points of X are precisely those having rank 0, points of rank 1 are those which are isolated in the subspace of all non-isolated points, and so on. For a non-empty С С X we define CBx(C) = sup{CBx(p) | p € C}; in this way CB^PO is defined and CBx({p}) = CBx(p) holds. If X is compact and С is closed in X then the sup is achieved: CBx(C) is the maximum value of CBx(p) for p € C; there are finitely many points of maximum rank in С and the number of such points is the CBx-degree of C, denoted by nx{C).
If X is countable and compact then CBx(^) is a countable ordinal and every closed subset has ordinal-valued rank and finite CB^-degree nx(x)euj\{ 0}.
For any ordinal a the set {p € X | CBx(p) > a} is called the a-th CB- derivative Xa of X.
Elements pel with CBx(p) = oo form the perfect kernel X^ of X.
Clearly, D Xa+i, a € Ord, and Xoo = f| X«.
a€Ord
Similarly, for a nontrivial superatomic Boolean algebra A the characteristics СВд(А), п_а(А), and СВд(р), for p € A, are defined [3] starting with atomic elements being isolated points. Following [3], СВд(А) and пд(А) are called the Cantor-Bendixson invariants, or CB-invariants of A.
Recall that by [3, Lemma 17.9], СВд(А) < for any infinite A, and the following theorem holds.
Theorem 2.7 [3, Theorem 17.11]. Countable superatomic Boolean algebras are isomorphic if and only if they have the same CB-invariants.
By Theorem 2.6 any e-totally transcendental family T defines a super-atomic Boolean algebra £>(T), and it is easy to observe step-by-step that RS(T) = СВв(г)(Б(Т)), ds(T) =nB{r)(B(T)), i.e., the pair (RS(T), ds(T)) consists of CB-invariants for B{T).
In particular, by Theorem 2.7, for any countable e-totally transcendental family T, £>(T) is uniquely defined, up to isomorphism, by the pair (RS(T),ds(T)) of CB-invariants.
By the definition for any e-totally transcendental family T each theory TeT obtains the CB-rank CB t(T) starting with T-isolated points To, of CBr(T0) = 0. We will denote the values CBr(T) by RSr(T) as the rank for the point T in the topological space on T which is defined with respect to E(T)-sentences.
3. Theories of abelian groups
Let A be an abelian group in the language E = (+1-2), — (^O^). Then kA denotes its subgroup {ka | a € A} and A[k] denotes the subgroup {a € A I ka = 0}. Let P be the set of all prime numbers. If p € P and pA = {0} then dim„4 denotes the dimension of the group A, considered as a vector space over a field with p elements. The following numbers, for arbitrary p € P and n € oj \ {0} are called the Szmielew invariants for the group A [2; 13]:
ap,n(A) = mm{dim((pnA)[p]/(pn+1A)[p]),io},
f3p(A) = min{inf{dim((yVl)[p] | n € oj},oj}, 7P(A) = min{inf{dim{{A/A[pn])/p{A/A[pn])) \ n € oj},oj}, e(A) € {0,1}, and e(A) = 0 (nA = {0}for somen € w,n / 0).
It is known [2, Theorem 8.4.10] that two abelian groups are elementary equivalent if and only if they have same Szmielew invariants. Besides, the following proposition holds.
Proposition 3.1 [2, Proposition 8.4.12]. Let for any p and n the cardinals av,n, fjp, tp < oj, and e € {0,1} be given. Then there is an abelian group A such that the Szmielew invariants aPin(A), f3p{A), 7P(A), ande(A) are equal to av,n, (3P, 7P, and e, respectively, if and only if the following conditions hold:
(1) if for prime p the set {n \ ap>n / 0} is infinite then [3P = 7P = oj;
(2) if e = 0 then for any prime p, [3P = 7P = 0 and the set {(p,n) | / 0} is finite.
We denote by Q the additive group of rational numbers, Zpn — the cyclic group of the order pn, Zp°o — the quasi-cyclic group of all complex roots of 1 of degrees pn for all n > 1, Rp — the group of irreducible fractions with denominators which are mutually prime with p. The groups Q, Zpn, Rp, Zpoo are called basic. Below the notations of these groups will be identified with their universes.
Since abelian groups with same Szmielew invariants have same theories, any abelian group A is elementary equivalent to a group
®p,n^-p"P' © ©p^pCO © (BpRp 9>®Q,{£), (3.1)
where B^ denotes the direct sum of k subgroups isomorphic to a group B. Thus, any theory of an abelian group has a model being a direct sum of based groups. The groups of form (3.1) are called standard.
Recall that any complete theory of an abelian group is based by the set of positive primitive formulas [2, Lemma 8.4.5], reduced to the set of the following formulas:
3y(mixi + ... + rrinxn &pky), (3.2)
m\X\ + ... + mnxn 0, (3.3)
where nii € Z, k € u, p is a prime number [1], [2, Lemma 8.4.7]. Formulas (3.2) and (3.3) allow to witness that Szmielew invariants defines theories of abelian groups modulo Proposition 3.1.
In view of Proposition 3.1 and equations (3.2) and (3.3) we have the following:
Remark 3.2. Theories of abelian groups are forced by sentences implied by formulas of form (3.2) and (3.3) and describing dimensions with respect to aPin, f3p, 7P, e as well as bounds for orders pk of elements and possibilities for divisions of elements by pk. Moreover, distinct values of Szmielew invariants are separated by some sentences modulo Proposition 3.1. Hence, counting ranks of families of theories of abelian groups it suffices to consider sentences separating Szmielew invariants.
4. Closures and ranks for families of theories of finite abelian
groups
Consider the family 7дйп of all theories of finite abelian groups. Clearly, Тл,An is countable corresponding to tuples of non-zero values of ap,n. By Proposition 3.1 the ^-closure of 7дAn produces theories, of infinite abelian groups, with some ap>n and [3P = 7P = oj. Since [3P = 7P = w can be obtained independently with respect to distinct p, we have |С1е(7дйп)| = 2Ш. Applying Theorem 2.3 we have:
Theorem 4.1. RS(71 fin J — OO.
Recall [4; 10] that an infinite structure Л4 is pseudofinite if every sentence true in Л4 has a finite model. Here the theory Th(7H) is also called pseudofinite.
Now we consider Szmielew invariants of theories in Cl£;(TA,fln)- Since theories of finite groups can not generate new theories of finite groups and finite abelian groups have finitely many nonzero values ap,n, and [3P = 7P = e = 0 for any prime p, it suffices to study theories of pseudofinite groups, i.e., theories in Ta,Pf = ClE(X4,fin) \ Ta,fin-
Notice by the way that by Theorem 4.1, RS(7A,pf) = since |TA,pf| = 2W in view of |ClE(7^)fln)| = 2W and |7^;fln| = u.
By Proposition 3.1 theories in 7a,pf are exhausted by limit values ap,n = uj, [3P = 7P = uj and e = 1 producing the following theorem.
Theorem 4.2. For any theory T of abelian groups the following conditions are equivalent:
(1) T € 7a,pf/
(2) T has some infinite ap,n, or some [3P = 7P = uj, or e = 1, moreover, for all nonzero values [3P and 7P, [3P = 7P = uj;
(3) T has infinite models, and all nonzero values [3P and 7P imply [3P = 7p = uj.
Proof. (1) (2). Let T € Ta,pf- Then T has infinite models and it is approximable by an e-minimal family T C Ta,fin, see [12, Proof of Theorem 6.1]. Therefore T is unique accumulation point for T. By the definition all theories in T has only finitely many positive values ap,n, all these values are natural numbers, and all Szmielew invariants f3p, 7P, e equal 0. Considering nonzero Szmielew invariants aP;il for theories in T, we have the following possibilities for each prime p:
(1) some value ap>n unboundedly increases for theories in T, with fixed
n;
(ii) ap,n / 0 with unboundedly many n, for theories in T;
(iii) values ap>n are bounded for theories in T, and {(p, n) \ ap>n / 0} is infinite for T.
In the case (i), T has ap>n = uj. In the case (ii), T has infinite {n \ Oip,n / 0} producing (3P = 7P = uj by Proposition 3.1, (1). In the case (iii), T has infinite {(p, n) \ ap>n / 0} producing e = 1 by Proposition 3.1, (2). Again by Proposition 3.1, T can not have finite positive [3P and 7p, and if /3P or 7p is positive then (3P = 7P = uj.
(2) ^ (1). Let T have some infinite CKP;ra, or some f3p = 7P = uj, or e = 1, and all positive values [3P and 7p imply (3P = 7P = uj. Now we construct step-by-step an e-minimal family T C Ta,fin of theories Tj = Th(^) of finite abelian groups Ai, i € uj, with unique accumulation point T, satisfying the following conditions:
a) for any i, Ai is a subgroup of Ai+i]
b) for any i-th prime number pi, if T has positive aPi,n, [3Pi, or 7Pi, then Aj, for j > i, have subgroups Z
^ i
c) if T has finite aP;il then the theories Tj have this Szmielew invariant starting with some i;
d) if T has infinite аР}П = аР}П(Т) then Tj have monotone increasing Szmielew invariants apn = aPtn(Ti) with lim apn(Ti) = w;
' ' i—>oo '
e) if T has [3P = 7P = w then either {n \ ap>n Ф 0} is infinite, for T, and all nonzero Szmielew invariants ap.n for Tj are exhausted by nonzero Szmielew invariants ap,n for T, or ap = {n \ ap>n Ф 0} is finite and Tj have distinct positive ар>п, n ^ ap;
f) if T has e = 1 then either {(p,n) | ap>n Ф 0} is infinite, for T, and all nonzero Szmielew invariants аР)П for Tj are exhausted by nonzero Szmielew invariants ap,n for T, or a = {(p, n) \ ap>n Ф 0} is finite and Tj have distinct positive аРуП, {p,n) £ a;
g) all nonzero Szmielew invariants ap n for Tj are described in the items b)-f).
The items c), d), g) guarantee the required invariants ap>n for the accumulation point T of T = {Ti | i € oj}, the items e), g) confirm the required invariants [3P and 7p for T, and the items f), g) — the value e of T. Thus,
T € TA>Pf.
(2) (3) immediately follows by Proposition 3.1. □
Notice that by Theorem 4.2 infinite standard groups
and, in particular, the group Q are pseudofinite.
Theorem 4.2 immediately implies:
Corollary 4.3. If a theory T of an abelian group has a positive natural value [3P or jp then models of T are not pseudofinite.
Since Th(Z) has values 7P = 1 [9], the group Z is not pseudofinite, as also noticed in [4].
Remark 4.4. Having Theorem 4.2 describing the set Та,pi we can study ranks for subfamilies T С Там and their closures Cle(T) С Там u Та,pi- By Theorem 4.1 these subfamilies T admit RS(T) = 00. Moreover, arguments for the proof of Theorem 4.1 show that one can choose 2Ш disjoint families T С Там RS(T) = 00. Indeed, we can define these families independently varying finite bounded values of some countably many ap>n staying infinitely many prime p' free for arbitrary values of ay;il. Therefore the values ap>n are responsible for 2Ш disjoint families T and api ,n are responsible for RS(T) = 00.
In view of Theorem 4.2 and Remark 4.4 we consider dynamics for RS(T) and ds(T), where T С Там■ The values RS(T) and ds(T) are defined by variations of possibilities for Szmielew invariants ap>n of T € T.
By Proposition 3.1 and Theorem 4.2 this dynamics can be realized by unbounded increasing of values of some ар>п, for fixed p and n, or by infinitely many nonzero аР}П, for (non-)fixed p and unbounded n.
For instance, taking TA,fm,p C Ta,fin consisting of theories of finite abelian groups whose positive Szmielew invariants are exhausted by aPtn, for chosen fixed p, we obtain 2W possibilities for Cl£;(7A,fin,p) varying independently for distinct n. Thus, RS(X4,fin,p) = oo.
Following Theorem 2.5, this rank value implies that 7a,fin,p has subfamilies T of arbitrary countable rank a and of arbitrary degree n. Below we show a mechanism to choose d-definable T with (RS(T),ds(T)) = (a,n).
5. e-Minimal families of theories and their accumulation points
In this section we consider both e-minimal families of theories of finite abelian groups and e-minimal subfamilies of the set 7a of all theories of abelian groups.
Let T be an e-minimal subfamily of Ta,fin- Then by the definition T has unique accumulation point T, models of T are pseudofinite, and Szmielew invariants are described in Theorem 4.2. It implies that, for T, some aPi,n is infinite, some [3P = 7P = u, or e = 1.
The value ap>n = uj is obtained by unbounded increasing sequence of finite ap,n = rrik for some theories Tk € T with these values. Thus,
®p,n = lim (Jp%, (5.1)
fc—S-oo
where ap,n is the value ap>n for T, aTkp,n are the values ap>n for Tk.
Following Proposition 3.1 the case [3P = 7P = uj corresponds infinitely many n with aP;il / 0 for T, i.e., by e-minimality, for infinitely many n there are infinitely many theories in T with aP;il / 0. Again by e-minimality it means that for each considered n there are finitely many theories in T with
(Xp,n — 0.
Again by Proposition 3.1 the case e = 1 implies infinitely many pairs (p,n) with ap,n / 0 for T, i.e., by e-minimality, for infinitely many (p,n) there are infinitely many theories in T with aP;il / 0, that is, for each considered (p, n) there are finitely many theories in T with ap>n = 0.
Similarly (5.1), in the latter two cases the values [3P = 7P = uj and/or e = 1 can be interpreted as limits lim oip n and/or lim ap n in the set of all
n ' p,n '
Szmielew invariants, where Szmielew invariants for some theories
in T. Since [3P = 7P € {0,uj} and e € {0,1} for any theory in CIe(T), we can assume that limo;pra € {0,w} and limo;pra € {0,1}.
n ' p,n '
Additionally, since e-minimal families have unique accumulation points, these limits are unique, too, i.e., there are no independent possibilities to obtain distinct limit values using distinct sequences of theories in T. In such a case we say that T does not have independent limit values.
It means that if we have some values for lim alkn, lima^, lima^ for
k—s-oo 1 ' n 1 ' p,n 1 '
an infinite sequence of theories T^ in T, we can not find another infinite sequence of theories in T producing different limit values.
Clearly, assuming that T does not have independent limit values we conversely obtain the e-minimality of T.
Thus, we have:
Theorem 5.1. For any infinite family T C Tam the following conditions are equivalent:
(1) T is e-minimal;
(2) T does not have independent limit values.
Considering an arbitrary infinite family T C Tam we define the finite or infinite number of independent limit values for T. This number is called the dimension of T and denoted by dim(T).
Remark 5.2. Let Tp,n be an infinite family consisting of theories T € 7a,fin with unbounded ap>n and fixed ay,n/ for (p',n') / (p,n). Clearly, dim(7^,ri) = 1. Besides,
for any finite set X of some pairs {p,n}. The equation (5.2) stays valid for infinite X if Tp,n are defined uniformly, say, if ay;il' = Const for (p', n') / (p,n). Otherwise, if fixed ay;il' are random, one can generate continuum
many pseudofinite theories T € CI® . Indeed, we can form
for some infinite and co-infinite set Z of prime numbers p such that ay;i, for theories in IJ TPii, are independently bounded and unbounded
(p,i)ex
for prime p' £ Z. Thus, there are continuum many possibilities for the
sequences of finite/infinite values ay;i for theories in Clg
Remind that here dim(T) is defined in terms of Szmielew invariants and their ordinal limits whereas e-Sp(T) is a model-theoretic value for the topology with respect to I?-closure.
Theorem 5.1 immediately implies the following reformulation:
Theorem 5.3. For any infinite family T C Tam the following conditions are equivalent:
(5.2)
Clearly, the value dim(T) equals the e-spectrum of T:
dim(T) = e-Sp(T).
(5.3)
(1) T is e-minimal;
(2) dim(T) = 1.
Remind [11] that e-minimality of T means that RS(T) = 1 and ds(T) = 1. Thus, these values for rank and degree are characterized by dim(T) = 1.
Similarly, finite dim(T) = n means that T has n accumulation points producing RS(T) = 1 and ds(T) = n as well as a representation of T as a disjoint union of n e-minimal subfamilies.
The following theorem gives an additional criterion for e-minimality of a family T C 7a,fin-
Theorem 5.4. An infinite family T C TA,fm of theories of abelian groups is e-minimal if and only if for any upper bound ap>n > m or lower bound ap>n < m, for rri there are finitely many theories in T satisfying this bound. Having finitely many theories with ap>n > m, there are infinitely many theories in T with a fixed value ap>n < m.
Proof. Let T be e-minimal. Consider a bound ap>n > m (respectively Oip>n < rn). By Remark 3.2 there is a sentence separating theories in T with Oip>n > rn (ap>n < m). Since T is e-minimal, exactly one of the conditions Oip>n > rn, Oip>n < m satisfies infinitely many theories in T. Thus, there are only finitely many theories in T satisfying (y,Ptn rn — 1 or (xPin ^ m. In the latter case, with infinitely many theories for aPi,n < m, the invariant aPtn should be repeated infinitely many times for distinct theories in T.
Conversely, we again apply Remark 3.2: taking an arbitrary sentence ip in the group language, we can describe only finitely many bounds ap>n > m and ctp,n ^ iTT" Since each bound forces finite or cofinite subfamily of T we have finite Tv or 7^,, i.e., T is e-minimal. □
Now we extend the context characterizing the e-minimality of subfamilies T in 7a-
Following Proposition 3.1 we again notice that all dependencies between values of Szmielew invariants in a given theory of an abelian group are exhausted by ones given by infinite {n \ ap>n /0} implying (3P = 7P = uj as well as by infinite {(p,n) | ap>n / 0} implying e = 1. It means that Szmielew invariants, for a fixed theory and for a family, can not force positive values aPiTl, f3p, jp using positive values for different prime p' and/or e. Besides, values ap>n and natural f3p, jp do not forced by other Szmielew invariants. Moreover, finite values apitl, f3p, 7P, for theories in C\e(T), can not be forced by other finite or infinite values of these invariants. Thus, all dependencies between distinct Szmielew invariants theories T e C\e(T) \T, are exhausted by the following ones for sequences (Tfc)fcew °f theories in T:
11 (T^1 = 1 im oP~'k
fc^-oo
2) $ = lim
3) 7J = lim 7PTfc,
r fc—S-oo r
3) eT = lim eTk,
k—>oo
4) Pp = lp = w = lim
5) eT = 1 = lim aLkn.
p,n 1 '
The items l)-5) show that limit values for Szmielew invariants are independent modulo apkn, i.e., the limits of /?Jfc, 7jfc, eTk can produce only 7J, eT, respectively, whereas dpkn can generate both = 7J = w
and eT = 1.
Thus, we can extend the notion of dimension dim(T) till arbitrary families T C 7a as the number of independent limit values for T. Notice that the equation (5.3) stays valid for the general case.
We denote by Szm the set {apitl \ p £ P,n £ uj \ {0}} U {(3p \ p € P}u{lp\peP}u {e}.
Theorem 5.5. For any infinite family T of theories of abelian groups the following conditions are equivalent:
(1) T is e-minimal;
(2) dim(r) = 1;
(3) for any upper bound { > m or lower bound £ < m, for rri € uj, of a Szmielew invariant { € Szm, there are finitely many theories in T satisfying this bound; having finitely many theories with £ > m, there are infinitely many theories in T with a fixed value ap>n <m, if { = ap,n, with a fixed value [3P < m, if £ = fip, with a fixed value 7v <m, if £ = 7P, and with a fixed value e < m, if£ = e.
Proof repeats arguments for Theorems 5.3 and 5.4 replacing ap>n by □
It is known [12, Theorem 7.3] that e-minimal families have unique accumulation points. Thus, Theorem 5.5 characterizes the possibilities of approximations of theories of abelian groups by families with unique accumulation points. Since theories of finite groups are isolated by complete sentences, the possibilities for the approximations are exhausted by theories of infinite groups. As the equations (3.2) and (3.3) can not produce complete sentences for infinite abelian groups, theories T of abelian groups A are approximable if and only if A are infinite. Moreover, in view of Proposition 3.1 these theories T can be approximated by e-minimal families as follows.
If T has finitely many prime p with positive Szmielew invariants and e = 0 then by Proposition 3.1, T is approximated by a family of theories Ti, i € uj, of finite groups and with fixed aP;il, if ap>n is finite for T, and with strictly increasing aP;il, if ap>n = uj for T.
If T has finitely many prime p with positive Szmielew invariants and £ = 1 then T is approximated by a family of theories Ti, i € uj, with £ = 1 and same positive values of Szmielew invariants for infinitely many
prime q, replacing p by q, including given p, and forming sets Qi such that Qi D Qi+1, i € u, and P| Qi consists of all p with positive Szmielew
i£ui
invariants for T.
If T has infinitely many prime p with positive Szmielew invariants then T is approximated by a family of theories Tj, i € u, with same positive values of Szmielew invariants for finitely many prime p, and forming sets Q\ such that Q\ C Q'i+i, i € w, and (J Q\ consists of all p with positive
i£ui
Szmielew invariants for T.
Thus, the following theorem holds.
Theorem 5.6. For any theory T of an abelian group A the following conditions are equivalent:
(1) T is approximated by some family of theories;
(2) T is approximated by some e-minimal family;
(3) A is infinite.
Remark 5.7. Theorem 5.6 characterizes accumulation points in the set of theories of abelian groups. Items l)-5) allows to define Szmielew invariants for accumulation points via correspondent Szmielew invariants of given theories. Thus, these items give possibilities to control theories of abelian groups by their approximations in terms of Szmielew invariants.
Remark 5.8. Theorems 4.2 and 5.6 describes accumulation points for the set of theories of finite abelian groups being theories of pseudofinite abelian groups. Thus the ranks for subfamilies T of Ta,fin are controlled by cardinalities and links in Cle(T) nTA,pf-
Now we can consider some particular possibilities for e-minimal approximations taking subfamilies T of the family Ta of all theories of abelian groups as follows.
If all Szmielew invariants except one of £ / e are fixed for theories in infinite TcTa then T is e-minimal with unique accumulation point having
Since there are continuum many sequences of Szmielew invariants / we have continuum many pairwise disjoint e-minimal subfamilies of Ta obtaining the following:
Proposition 5.9. The family Ta contains continuum many pairwise disjoint e-minimal subfamilies.
Again, in particular, if £ / e is unique nonzero Szmielew invariant for an infinite T C Ta then T is e-minimal whose unique accumulation point has exactly one nonzero Szmielew invariant £ = uj modulo e. And there are countably many such e-minimal families.
6. Ranks for families of theories
Since there are continuum many theories of abelian groups, varying Szmielew invariants, Theorem 2.3, as well as Theorem 4.1 and monotony of rank imply the following proposition for the family Та-
Proposition 6.1. RS(7a) = oo.
Having RS = oo for the family of all theories of (finite or infinite) abelian groups, we also notice that there are continuum many theories of divisible or, respectively, torsion free abelian groups that again produces RS = oo for the families of all theories of divisible/torsion free abelian groups.
By Theorem 2.4 and 4.1 there are many e-minimal d-definable subfamilies of Та- Below we generalize Theorem 2.4 for Та and arbitrary pair (a, n), where a is a countable ordinal and n € w \ {0}.
Theorem 6.2. Let a be at most countable ordinal, n € w\ {0}. Then there is a d-definable subfamily (Та)ф such that RS((7a)#) = ol and ds((X4b) = n.
Proof. If a < 1 we can apply arguments for [6, Theorem 4.6] obtaining the assertion. If a > 2 we consider a countable family of countable subsets of the set p, where A is a sequence of indexes for some ¡3 < a, i £ oj, as follows. Take an arbitrary countable family T С Та with RS(T) = a and ds(T) = n. By [6, Proposition 4.4] there are countably many countable s-definable subfamilies Tp with RS(7^>) = (3 < a, (3 > 0, and ds(7^>) = 1 witnessing RS(T) = a and ds(T) = n such that distinct subfamilies with RS = (3 are disjoint. We can code these subfamilies by where A is a sequence coding the rank of taken subfamily as well as containing codes of all tp containing i g oj. Now we replace by countable Хгл с p such that:
1) if A witnesses a rank (3 then Хгл П XA = 0 for i ф j;
2) if A and A' witness ranks (3 and 7, (3 < 7, and A contains the code for then XA с ХгА for each j, moreover, ХгА = IJ where the
union is taken with respect to all admissible codes A, for the rank [3 and inclusions с ХгА , and indexes j enumerating families
We denote by Y the union of all sets and assume that P \ Y is infinite. Prime numbers in P\Y are called free and will be used to mark s-definable subfamilies of theories in Та- Now we construct, in the following way, the required subfamily (Та)ф consisting of theories in families (7a)a correspondent to the families and satisfying the conditions:
1') if A' and A" witness a rank ¡3 then (Ti)f П {TA)f" = 0 for i ф j;
2') if A' and A" witness ranks (3 and 7, (3 < 7, and A' contains the code
(A",i) then (TA)f С СTA)f for each j, moreover, ('TA)f = U (TA)f -
A'j
We assume that all theories in (7а)ф satisfy cyi € {0,1}, p € P, Oip>n = 0, for n > 2 and p € P, [3P = 7P = 0 for p € P, e = 1. If A codes the rank 1 then Хгл defines the e-minimal, in view of Theorem 5.5, family (Та)?' of all theories T of abelian groups with unique positive ap>i, where p € Хгл, and with some positive суд for (3 free prime numbers p' marking Хгл and all X^ containing Хгл such that writing some суд = 1 we can separate (Та)? both from bigger families (TClf, collecting e-minimal families (Та)?" with суд = 1, and RS((7a)^') > ^((Ta)^) and from distinct marked families of the same rank. Now we unite (Та)? by values суд € {0,1} forming the families (7a)^', and the required indefinable subfamily T = Cle U (Та)? of 7a satisfies RS(T) = a and
\А,г J
ds(T) =n.a
Remark 6.3. Varying values cyra for the families in the construction for the proof of Theorem 6.2 we obtain continuum many disjoint, modulo e, families (7а)ф such that RS((7a)$) = a and ds((7A)$) = n.
Notice that the arguments stay valid replacing cyra by [3P or/and 7p.
Applying the construction for the proof of Theorem 6.2 we can form a family T of theories of finite abelian groups for given rank and degree, with a d-definable closure Cle(7~) с 7а;ап U7A,pf- Thus, the following theorem holds:
Theorem 6.4. Let a be a countable ordinal, n € w \ {0}. Then there is a subfamily T С Там such tha,t RS(7~) = a, ds(T) = n, and CIe(T) С 7a,fln U 7a,pf is d-definable with (RS(Cle(T)), ds(Cle(T)) = (a,n).
Remark 6.5. Theorems 2.6, 2.7, and 6.2 allow to form countable superatomic Boolean algebras Ва>п, unique up to isomorphism, for Inclosed, d-definable families of abelian groups with arbitrary countable CB-invariants (a,n). Additionally, Theorem 6.4 produces similar realizations using Ш-closed, d-definable subsets of 7a,ап U 7a,pf.
7. Conclusion
We found ranks for families of theories of abelian groups. In particular, we studied closures and ranks for theories of finite abelian groups observing that the set of theories of finite abelian groups in not totally transcendental. We characterized pseudofinite abelian groups in terms of Szmielew invariants, e-minimal families of theories of abelian groups are characterized both in terms of dimension and in terms of inequalities for Szmielew invariants. These characterizations were obtained both for finite abelian groups and in general case. Furthermore we gave characterizations for approximability of theories of abelian groups and show the possibility
to count Szmielew invariants via these parameters for approximations. We described possibilities to form d-definable families of theories of abelian groups having given countable rank and degree.
References
1. Eklof P.C., Fischer E.R. The elementary theory of abelian groups. Annals of Mathematical Logic, 1972, vol. 4, pp. 115-171. https://doi.org/10.1016/0003-4843(72)90013-7
2. Ershov Yu.L., Palyutin E.A. Mathematical logic. Moscow, Fizmatlit Publ., 2011.
3. Koppelberg S. Handbook of Boolean Algebras. Vol. 1, Monk J.D., Bonnet R. (eds.). Amsterdam, New York, Oxford, Tokyo, North-Holland, 1989, 342 p.
4. Macpherson D. Model theory of finite and pseudofinite groups. Archive for Mathematical Logic, 2018, vol. 57, no. 1-2, pp. 159-184. https://doi.org/10.1007/s00153-017-0584-1
5. Markhabatov N.D., Sudoplatov S.V. Ranks for families of all theories of given languages. arXiv: 1901.09903vl [math.LO], 2019, 9 p.
6. Markhabatov N.D., Sudoplatov S.V. Definable subfamilies of theories and related calculi. arXiv: 1901.08961vl [math.LO], 2019, 20 p.
7. Morley M. Categoricity in Power. Transactions of the American Mathematical Society, 1965, vol. 114, no. 2, pp. 514-538. https://doi.org/10.2307/1994188
8. Pavlyuk In.I., Sudoplatov S.V. Families of theories of abelian groups and their closures Bulletin of Karaganda University. Ma,them,a,tics, 2018, vol. 92, no. 4, pp. 72-78. https://doi.org/10.31489/2018M4/72-78
9. Popkov R.A. Distribution of countable models for the theory of the group of integers. Siberian. Math. J., 2015, vol. 56, no. 1, pp. 185-191. https://doi.org/10.1134/S0037446615010152
10. Rosen E. Some Aspects of Model Theory and Finite Structures. The Bulletin of Symbolic Logic, 2002, vol. 8, no. 3, pp. 380-403. https://doi.org/10.2307/3062205
11. Sudoplatov S. V. Ranks for families of theories and their spec-tra.arXiv: 1901.08464vl [math.LO], 2019, 17 p.
12. Sudoplatov S.V. Approximations of theories. arXiv: 1901.08961vl [math.LO], 2019, 16 p.
13. Szmielew W. Elementary properties of Abelian groups. Fund. Math., 1955, vol. 41, pp. 203-271. https://doi.org/10.4064/fm-41-2-203-271
Inessa Pavlyuk, Candidate of Sciences (Physics and Mathematics); Associate Professor of Chair of Informatics and Discrete Mathematics, Novosibirsk State Pedagogical University, 28, Vilyuiskaya st., Novosibirsk, 630126, Russian Federation, tel.: (383)2441586 (e-mail: inessa7772@mail.ru)
Sergey Sudoplatov, 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 25.04.19
Ранги семейств теорий абелевых групп
Ин. И. Павлюк
Новосибирский государственный педагогический университет, Новосибирск, Российская Федерация
С. В. Судоплатов
Институт математики им. С. Л. Соболева СО РАН, Новосибирский государственный технический университет, Новосибирский государственный университет, Новосибирск, Российская Федерация
Аннотация. Ранг семейства теорий подобен рангу Морли и может служить мерой сложности или богатства данного семейства. Увеличивая ранг расширениями семейства, мы получаем более богатые семейства, которые в случае достижения бесконечности могут рассматриваться как "достаточно богатые". В данной статье реализуются ранги для семейств теорий абелевых групп. В частности, изучаются ранги и замыкания для семейств теорий конечных абелевых групп. Показано, что множество теорий конечных абелевых групп не является тотально трансцендентным, т.е. его ранг равен бесконечности. В терминах шмелевских инвариантов характеризуются псевдоконечные абелевы группы. Кроме того, характеризуются е-минимальные семейства теорий абелевых групп как на языке размерности, т.е. числа независимых пределов шмелевских инвариантов, так и в терминах неравенств для шмелевских инвариантов. Эти характеризадии получены для конечных абелевых групп и в общем случае. Найдены характеризадии аппроксимируемости теорий абелевых групп и показаны возможности подсчета шмелевских инвариантов через параметры аппроксимаций. Описаны возможности построения (¿-определимых семейств теорий абелевых групп, имеющих данный счетный ранг и данную степень.
Ключевые слова: семейство теорий, абелева группа, ранг, степень, замыкание.
Список литературы
1. Eklof Р. С., Fischer Е. R. The elementary theory of abelian groups // Annals of Mathematical Logic. 1972. Vol. 4. P. 115-171. https://doi.org/10.1016/0003-4843(72)90013-7
2. Ершов Ю. Л., Палютин E. А. Математическая логика. M. : Физматлит, 2011.
3. Koppelberg S. Handbook of Boolean Algebras. Vol. 1 / eds. J. D. Monk, R. Bonnet. Amsterdam, New York, Oxford, Tokyo : North-Holland, 1989. 342 p.
4. Macpherson D. Model theory of finite and pseudofinite groups // Archive for Mathematical Logic. 2018. Vol. 57. N 1-2. P. 159-184. https://doi.org/10.1007/s00153-017-0584-l
5. Markhabatov N. D., Sudoplatov S. V. Ranks for families of all theories of given languages // arXiv:1901.09903vl [math.LO], 2019. 9 p.
6. Markhabatov N. D., Sudoplatov S. V. Definable subfamilies of theories and related calculi // arXiv: 1901.08961vl [math.LO], 2019. 20 p.
7. Morley M. Categoricity in Power // Transactions of the American Mathematical Society. 1965. Vol. 114, N 2. P. 514-538. https://doi.org/10.2307/1994188
8. Pavlyuk In. I., Sudoplatov S. V. Families of theories of abelian groups and their closures // Bulletin of Karaganda University. Mathematics. 2018. Vol. 92, N 4. P. 72-78. https://doi.org/10.31489/2018M4/72-78
9. Popkov R. A. Distribution of countable models for the theory of the group of integers // Siberian. Math. J. 2015. Vol. 56, N 1. P. 185-191. https://doi.org/10.1134/S0037446615010152
10. Rosen E. Some Aspects of Model Theory and Finite Structures // The Bulletin of Symbolic Logic. 2002. Vol. 8, N 3. P. 380-403. https://doi.org/10.2307/3062205
11. Sudoplatov S. V. Ranks for families of theories and their spectra // arXiv:1901.08464vl [math.LO], 2019. 17 p.
12. Sudoplatov S. V. Approximations of theories // arXiv:1901.08961vl [math.LO]. 2019. 16 p.
13. Szmielew W. Elementary properties of Abelian groups // Fund. Math. 1955. Vol. 41. P. 203-271. https://doi.org/10.4064/fm-41-2-203-271
Инесса Ивановна Павлюк, кандидат физико-математических наук, доцент кафедры информатики и дискретной математики, Новосибирский государственный педагогический университет, 630126, Российская Федерация, г. Новосибирск, ул. Вилюйская, 28, тел. (383)2441586 (e-mail: inessa7772@mail.ru)
Сергей Владимирович Судоплатов, доктор физико-математических наук, доцент, ведущий научный сотрудник, Институт математики им. С. JI. Соболева СО РАН, 630090, Российская Федерация, г. Новосибирск, пр. Академика Коптюга, 4, тел.: (383)3297586; заведующий кафедрой алгебры и математической логики, Новосибирский государственный технический университет, 630073, г. Новосибирск, пр. К. Маркса, 20, Российская Федерация, тел. (383)3461166; профессор кафедры алгебры и математической логики, Новосибирский государственный университет, 630090, г. Новосибирск, ул. Пирогова, 1, Российская Федерация, тел. (383)3634020 (e-mail: sudoplat@math.nsc.ru)
Поступила в редакцию 25.04-19
