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Серия «Математика»
2019. Т. 28. С. 85-94
УДК 510.67:512.577
MSG 03С15, 03С30, 03С50
DOI https://doi.org/10.26516/1997-7670.2019.28.85
Ranks for Families of Permutation Theories *
N. D. Markhabatov
Novosibirsk State Technical University, Novosibirsk, Russian Federation
Abstract. The notion of rank for families of theories, similar to Morley rank for fixed theories, serves as a measure of complexity for given families. There arises a natural problem of describing a rank hierarchy for a series of families of theories.
In this article, we answer the question posed and describe the ranks and degrees for families of theories of permutations with different numbers of cycles of a certain length. A number examples of families of permutation theories that have a finite rank are given, and it is constructed a family of permutation theories having a specified countable rank and degree n. It is proved that in the family of permutation theories any theory equals a theory of a finite structure or it is approximated by finite structures, i.e. any permutation theory on an infinite set is pseudofinite. Topological properties of the families under consideration were studied.
Keywords: family of theories, pseudofinite theory, permutation, rank, degree.
A rank for the families of theories, similar to Morley rank and defined in [9], 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 obtaining families with the infinite rank that can be considered "rich enough".
Permutation theories and theories in the language of one unary function have been studied in a number of papers, including [1; 2; 5; 7; 8]. In the present paper, we describe ranks and degrees for families of permutation theories, partially answering a question in [9].
* This research was partially supported by Committee of Science in Education and Science Ministry of the Republic of Kazakhstan (Grants No. AP05132349) and Russian Foundation for Basic Researches (Project No. 17-01-00531-a).
1. Preliminaries
Throughout hereinafter we consider families T of complete first-order theories of a language E = E(T) and use the following terminology from [3;6;9; 10].
Definition 1. [10] Let T be a family of theories and T be a theory, T^T. The theory T is called T-approximated, or approximated by T, or T - approximable, or a pseudo-T -theory, if for any formula tp € T there is T'er such that <p € T'.
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.
We put Tp = {T e T \ (p G T}. Any set Tp is called the tp-neighbourhood, or simply a neighbourhood, for T.
An approximating family T is called e-minimal if for any sentence cp € E(T), Tp is finite or T^ is finite.
It was shown in [10] that any e-minimal family T has unique accumulation point T with respect to neighbourhoods Tp, and Tu {T} is also called e-minimal.
Proposition 1. [10] A theory T T is T-approximated if and only if TeClE(T).
Definition 2. [6] An infinite structure A4 is pseudofinite if every sentence true in A4 has a finite model.
If T = Th(M) for pseudofinite M then T is called pseudofinite as well. We denote by T the class of all complete elementary theories, by T/m the subclass of T consisting of all theories with finite models.
Proposition 2. [10] For any theory T the following conditions are equivalent:
(1) T is pseudofinite;
(2) T is Tfin-approximated;
(3) T € CIe{Tfin)\Tfin-
Following [9] we define the rank RS(-) for the families of theories, similar to Morley rank [4], and a hierarchy with respect to these ranks in the following way.
RS(T) = — 1, if the family T is empty;
RS(T) = 0, if the family T is finite and non-empty;
RS(T) >1, if the family T is infinite.
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 € u, such that RS(7^n) > /3, n € uj.
If a is a limit ordinal then RS(T) > a if RS(T) > fi for any fi < 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 3. [9] If an infinite family T does not have e-minimal subfamilies Tv then T is not totally transcendental.
If T is e-totally transcendental, with RS(T) = a > 0, we define the degree ds(T) of T as the maximal number of pairwise inconsistent sentences cpi such that RS(7^) = a.
It is described in [3] the ranks and degrees for the families 7s of all theories of an arbitrarily given language E and (non-) totally transcendental families were characterized.
Theorem 1. [3] //E is a language containing an m-ary predicate symbol, for m >2, or an n-ary functional symbol, for n > 1, then RS(7s) = oo.
Furthermore, it is shown an applications of these characteristics for the families 7s)ra of all theories of languages E and having n-element models, where new, as well as for the 7s)0o families of all theories of languages E and having infinite models.
Clearly, for any language E, 7s = U Tk,n U 7s)0o- Therefore, by
n£ui
monotony of RS, we have for any new the following relations are true:
RS(7s,„) < RS(Ts), RS(7s,oo) < RS(7s).
Theorem 2. [3] For any language E either RS(7s)ra) = 0, if E is finite or n = 1 and E has finitely many predicate symbols, or RS(7s)ra) = oo, otherwise.
Theorem 3. [3] For any language E either RS(7s;0o) is finite, if E is finite and without predicate symbols of arities m > 2 as well as without functional symbols of arities n > 1, or RS(7s;0o) = oo, otherwise.
Proposition 4. [10] Any family T of theories can be expanded till a family T' with the least generating set.
Definition 3. [9] A family T, with infinitely many accumulation points, is called a-minimal if for any sentence (p e E(T), %p or T^ has finitely many accumulation points.
Let a be an ordinal. A family T of rank a is called a-minimal if for any sentence (p e E(T), RS(7^,) < a or RS(7^) < a.
Proposition 5. [9] (1) A family T is 0-minimal if and only if T is a singleton.
(2) A family T is I-minimal if and only if T is e-minimal.
(3) A family T is 2-minimal if and only ifT is a-minimal.
(4) For any ordinal a a family T is a-minimal if and only ¿/RS(T) = a and ds(T) = 1.
Theorem 4. [9] For any family T, RS(T) = 2 with ds(T) = n, if and only ifT is represented as a disjoint union of subfamilies Tifl,..., Tpn, for some pairwise inconsistent sentences <pi,..., <pn, such that each Tt'Pi is a-minimal.
Proposition 6. [9] For any family T, RS(T) = a with ds(T) = n, if and only ifT is represented as a disjoint union of subfamilies TPl,... ,Tpn, for some pairwise inconsistent sentences <pi,..., <pn, such that each Tt'Pi is a-minimal.
2. Ranks for families of permutation theories
In this section, we describe the ranks for families of permutation theories.
Let a language E consist of the permutation /. Denote by 7s the family of all permutation theories of language E.
Each permutation / has the axiom \/y3=1x(f(x) = у). The length of a permutation cycle is the number of its elements. The type of permutation / is the vector A(/) = (Ai(/),..., Лn(f ), ■ • •)> where Л¿(/) is the number of cycles of length i in the permutation /. Note that for any permutation / the value Ym=i ^' is equal to the power of the set of elements that make up the cycles.
Let T be a permutation theory, Л4 |= T. An element a € M is called acyclic if a does not belong to any cycle.
Note that by the compactness theorem the theories with cycles of unbounded length generate acyclic elements in some their models. If the cycle lengths are limited in aggregate, then there are theories with both acyclic elements and without acyclic elements.
For a given theory of T permutations, we can consider the set of pairs (n, Xn), where new and Xn € w U {oo} is the number of cycles of length n. As e € {0,1}, indicating the absence / presence of successor function, we can take the value 0 if there is no successor function and 1 if there is such a function. Moreover, e = 1 if {Ara > 0 | n € w} is infinite. In particular, the closure contains the theory with e = 1, if there are theories in this family with Xn > 0 for an infinite number of values of n.
Note that the characteristics (n, Xn), n € w, and e uniquely define this permutation theory. The ranks of families of permutation theories are given by sets of these characteristics. Below we consider all possible cases.
Families with a bounded number of positive Xn with e = 0
A family T C 7s of permutations can be infinite only if there are unbounded values of Xn, and if the family has a finite number of variants for Xn, then it is finite, RS(Ts) = 0 and ds(7s) is equal to the number of these variants; if Xn has an infinite number of variants in the theories from this family, then we need to look at the number of accumulation points, with Xn = oo, depending on whether the remaining Xn are fixed or not. If the number of accumulation points is finite, then RS(7s) = 1 and ds(7s) correspond to the number of these accumulation points. If the number of accumulation points is infinite, then RS(7s) > 2 and we need to look at how many accumulation points the accumulation points themselves generate.
Example 1. Let 7^ be the family for all identical permutations. Then RS(T-g) = 1 and ds(T^) = 1, therefore, 7^ is e-minimal. The only accumulation point is the theory of identical permutations on an infinite set.
Example 2. If we consider the cycles of length no and n\ and the number of these cycles satisfies Xno < k,Xni < I, then for the family T\no,\ni of theories with these relations, we have k ■ I variants, RS(T\ \ ) = 0, and ds(T\„0,\ni) = k ■ I.
We denote by Tn the set of all theories from 7s with one arbitrary value Xn, where Xm = 0 for m / n and the models of these theories do not have acyclic elements.
Proposition 7. Each family Tn is e-minimal.
Proof. The family Tn consists of theories Tm with m € w\{0} cycles of length n and the theory T,'x with an infinite number of cycles of length n. The theory T^ is the unique accumulation point for Tn- Thus, RS(Tn) = 1, ds(Tn) = 1 and therefore the family Tn is e-minimal. □
Example 3. If we allow cycles of different lengths no and n\, then we get a countable number of variants (Arao, Arai), where Xno is the number of cycles of length no, and Xni is the number of cycles of length n\. Thus, there are countably many theories with cycles of length no and n\, forming the family Tn0,ni■ Here, each theory with one infinite Xno or Xni has RS(Tno,ni) = 1, and the only limit point c Xno = Xni = oo, has infinitely many cycles of length no, infinitely many cycles of length n\ and RS(Tno,ni) = 2. Thus, for a given family Tno,ni, we obtain RS(Tn0,ni) = 2 and ds(Tno,ni) = 1-Therefore, the family is a-minimal.
Example 4. If we consider cycles of different lengths no,n\ and «2, then you also obtain countably many possibilities (Xno, Xni, Xn2), where Xno is the number of cycles of length no, Xni is the number of cycles of length n\, Xn2 is the number of cycles of length «2. Each e-minimal subfamily
with one infinite Xno, Xni or Xm containing theories with only one positive Xni has the rank RS = 1. Theories with nonzero Xni, Xnj and АПк = 0, {i,j,k} = {0,1,2} have RS = 2 and ds = 1. A family of all theories with Хщ = 0 for m (ji {п0,п\,п2} has RS = 3 and ds = 1.
Thus, adding new Xn for cycles of a certain length, one can unboundedly increase the rank to any pre-given natural number.
Families with a bounded number of positive Xn with e = 1
As in the previous case, families are e-totally transcendental and may contain e-minimal, a-minimal, a-minimal subfamilies. As well as their model may contain copies of successor function (Z, s) on integers. In this case, the rank of a family is ordinal. Repeating the argument for the previous case, we obtain the ranks RS = a and ds = m.
By the definition of a-minimality and Proposition 7, the family T of theories of permutations with RS{T) = a and ds(T) = m can be represented as a disjoint union of subfamilies T\„ ,..., T\„ , for some different
о riQ I I nm — 1 '
Xno,..., ХПт_1, such that each T\i is a-minimal.
The following theorem shows that there is a family of permutation theories, having a countable rank.
Theorem 5. For any countable ordinal a and natural к > 1 there exists a family T С 7s, such that RS{T) = a and ds{T) = k.
Proof. Realizations of finite ranks by families of permutation theories shows that, in order to prove the theorem, it suffices to construct a family of theories that has a specified countable rank and degree n.
We choose the number s € w \ {0} and let the considered theories of permutations T have an arbitrary number of cycles of length s and at most one cycle of length m for each m Ф s.
This countable rank a for a family of permutation theories T can be constructed using countable sets X consisting of natural numbers m ф s, each of which symbolizes the presence of a single cycle of length m in models of theory T and the absence of cycles of length m' ф s with m' £ X. Since the equality RS = a implies that the corresponding Boolean algebra is superatomic, the families X must form a hierarchy in which every transition from the set X, specifying a family of rank (3 < a, to the sets Xj, i € w, specifying disjoint subfamilies of theories of lower rank, must satisfy the following conditions:
X сХг,\Хг\Х\=ш, (Хг П Х3) \ X = 0,
where i,j € oj,i ф j.
Herewith the chains with respect to the inclusion consisting of the sets X should be well ordered. The sets X are indexed by the pairs ((3,k),
where 5 < a, k < m for 5 = a, and k € w for 5 < a. Thus, the sets Xg for the ordinal 5 = 7 + 1 are expanded by a countable family of sets X(^,k) pairwise disjoint over Xg.
Each set X(1;fc) defines a suitable e-minimal family of 7(1;fc) theories having one cycle of each length m € and an arbitrary number of
cycles of length s. Denote by T the union of all families 7^). Using induction, it is easy to show that the formulas ^g, describing the presence of cycles of length m' € (5, k'), they define neighborhoods of 7^ with rank /3. Thus, RS(7~) = a and ds(T) = n is established. □
Families with infinitely many positive An
In the family 7s of permutation theories there are theories with infinite cycles and cycles with unbounded lengths. As well as in the models of these theories we observe automatically the presence of a copy of successor function (Z, s) on integers. In this case, there is an infinite 2-tree formed thus, the family has RS(7s) =
Theorem 6. Any theory T of a permutation on an infinite set is pseud-ofinite.
Proof. Case 1. Let T have a finite number of cycles. Then T has a model M = M0 U M1, where M0 is a subsystem consisting of cycles, and M1 is a subsystem without cycles. For this model, the following is true:
M = lim Mi,
i—
where Mi = M0 U Ni is finite and Ni is the structure consisting of one cycle of length i. Thus {Th(Ni) | i € w} approximates the theory Th(M1), and {Th(Mi) | i € w} approximates the theory of T.
Case 2. Let the theory T have infinitely many cycles and the lengths of the cycles are bounded in the aggregate. Let n0,...,nk be the cycle lengths, A0,..., Afc are the number of cycles of length n0,..., nk, A0,..., Ar finite, Ar+1,..., Ak are infinite, r < k and M0 U Ni, where Ni consists of i cycles of length nr+1,..., nk. Then the set {Th(Ni) | r € w} approximates the theory Th(M1), where M1 consists of an infinite number of cycles of length nr+1,... ,nk. Thus, {Th(M0 uNi)|i € w} approximates the theory T = Th(M0 UNU).
Case 3. Let T have infinitely many cycles and the lengths of the cycles are not limited in aggregate. Let n0,..., nk,... be the cycle lengths. In this case, the theory is T = Th(M0 UM1), where M0 is a subsystem consisting of cycles, and M1 is a subsystem without cycles. The Ni subsystem consists of < i cycles of length j < i and does not contain cycles of length > i. Let ns be the cycle length, As be the number of cycles of length ns in the model of the theory T. Then Ni contains min{i, As} cycles of length ns. The subsystem M1 is obtained by the compactness theorem. So the T theory is approximated by a family of theories Th(Ni). □
Since -E-closures of families of theories preserve the rank [9, Theorem 2.10], Theorem 6 shows that to calculate the ranks of families of permutation theories, it suffices to consider suitable families of permutation theories on finite sets.
3. Conclusion
In the paper the ranks and degrees for families of permutation theories with different numbers of cycles of a certain length are described. Several examples of families of finite rank permutation theories are given. A family of permutation theories is constructed that has a specified countable rank and degree n. It is proved that any permutation theory on an infinite set is pseudofinite. Topological properties of families of permutation theories are studied.
References
1. Ivanov A.A. Complete theories of unars. Algebra and Logic, 1984, vol. 23, no. 1, pp. 36-55.
2. Marcus L. The number of countable models of a theory of one unary function. Fundamenta Mathematicae, 1980, vol. CVIII, issue 3, pp. 171-181.
3. Markhabatov N.D., Sudoplatov S.V. Ranks for families of all theories of given languages. arXiv:1901.09903vl [math.LO], 2019, available at: https://arxiv.org/abs/1901.09903vl
4. Morley M. Categoricity in Power. Transactions of the Am,erican Mathematical Society, 1965, vol. 114, no. 2, pp. 514-538.
5. Popkov R.A. Klassifikacija schjotnyh modelej polnyh teorij odnomestnyh predika-tov s podstanovkoj ogranichennogo porjadka [Classification of countable models of complete theories of unary predicates with permutation of bounded order], Algebra and Model Theory 8. Collection of papers, NSTTJ, Novosibirsk, 2011, pp. 73-82.
6. 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/10.2178/bsl/1182353894
7. Ryaskin A.N. The number of models of complete theories of unars. Model Theory and Its Applications, Tr. Inst. Mat. SO AN SSSR, 1988, vol. 8, pp. 162-182.
8. Shishmarev Yu.E. On categorical theories of one function. Mat. Zametki, 1972, vol. 11, no. 1, pp. 89-98.
9. Sudoplatov S.V. Ranks for families of theories and their spectra. arXiv: 1901.08464vl [math.LO], 2019,
available at: https://arxiv.org/abs/1901.08464 10. Sudoplatov S.V. Approximations of theories. arXiv:1901.08961vl [math.LO], 2019, available at: https://arxiv.org/abs/1901.08961
Nurlan Markhabatov, Postgraduate, Novosibirsk State Technical University, 20, K. Marx Ave., Novosibirsk, 630073, Russian Federation, tel.: (383)3461166; (e-mail: nur_24.08.93@mail.ru)
Received 25.04.19
Ранги для семейств теорий подстановок
Н. Д. Мархабатов
Новосибирский государственный технический университет, Новосибирск, Российская Федерация
Аннотация. Понятие ранга для семейств теорий, аналогичное рангу Морли для фиксированных теорий, служит мерой сложности для данных семейств. Возникает естественная проблема описания иерархии ранга для ряда семейств теорий.
В данной статье мы, отвечая на поставленный вопрос, описываем ранги и степени для семейств теорий подстановок с разным числом циклов определенной длины. Приведено несколько примеров семейств теорий подстановок, которые имеют конечный ранг, а также построено семейство теорий подстановок, имеющее данный счетный ранг и данную степень те. Доказано, что в семействе теорий подстановок любая теория является теорией конечной структуры или аппроксимируется теориями конечных структур, т. е. любая теория подстановки на бесконечном множестве является псевдоконечной. Изучены топологические свойства рассматриваемых семейств.
Ключевые слова: семейство теорий, псевдоконечная теория, подстановки, ранг, степень.
Список литературы
1. Ivanov A. A. Complete theories of unars // Algebra and Logic. 1984. Vol. 23, N 1. P. 36-55.
2. Marcus L. The number of countable models of a theory of one unary function // Fundamenta Mathematicae. 1980. Vol. CVIII, Issue 3. P. 171-181.
3. Markhabatov N. D., Sudoplatov S. V. Ranks for families of all theories of given languages // arXiv:1901.09903vl [math.LO], 2019, available at: https://arxiv.org/abs/1901.09903vl
4. Morley M. Categoricity in Power // Transactions of the American Mathematical Society. 1965. Vol. 114, N 2. P. 514-538.
5. Popkov R. A. Klassifikacija schjotnyh modelej polnyh teorij odnomestnyh predikatov s podstanovkoj ogranichennogo porjadka [Classification of countable models of complete theories of unary predicates with permutation of bounded order] // Algebra and Model Theory 8. Collection of papers / NSTU. Novosibirsk, 2011. P. 73-82.
6. 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/10.2178/bsl/1182353894
7. Ryaskin A. N. The number of models of complete theories of unars // Model Theory and Its Applications, Tr. Inst. Mat. SO AN SSS. 1988. Vol. 8. P. 162-182.
8. Shishmarev Yu. E. On categorical theories of one function // Mat. Zametki. 1972. Vol. 11, N 1. P. 89-98.
9. Sudoplatov S. V. Ranks for families of theories and their spectra // arXiv: 1901. 08464vl [math.LO], 2019. Available at: https://arxiv.org/abs/1901.08464
10. Sudoplatov S. V. Approximations of theories // arXiv:1901.08961vl [math.LO], 2019. Available at: https://arxiv.org/abs/1901.08961
Нурлан Дарханович Мархабатов, аспирант, Новосибирский государственный технический университет, Российская Федерация, 630073, г. Новосибирск, проспект К. Маркса, 20, тел. (383)3461166; (e-mail: nur_24.08.93@mail.ru)
Поступила в редакцию 25.04-19
