A Certain Reducibility on Admissible Sets

Siberian Mathematical Journal, Vol. 50, No. 2, pp. 330–340, 2009 c 2009 Puzarenko V. G. Original Russian Text Copyright ...

Author: Puzarenko V.G.

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Siberian Mathematical Journal, Vol. 50, No. 2, pp. 330–340, 2009 c 2009 Puzarenko V. G. Original Russian Text Copyright

A CERTAIN REDUCIBILITY ON ADMISSIBLE SETS V. G. Puzarenko

UDC 510.5:510.225

Abstract: We introduce a certain reducibility on admissible sets which preserves deﬁnable predicates. Some lattice-theoretic properties are given of the ordered sets of the classes of admissible sets equivalent under this reducibility. Furthermore, we give a transformation that assigns to each admissible set some hereditarily ﬁnite set and preserves the following list of descriptive set-theoretic properties (with account taken of the levels of a deﬁnable hierarchy): enumerability, quasiprojectibility, uniformization, existence of a universal function, separation, and extension. We introduce the notion of jump of an admissible set which translates the descriptive set-theoretic properties considered above to the corresponding properties lowering levels by 1. Keywords: computably enumerable set, enumeration reducibility, Σ-reducibility, descriptive set-theoretic properties, admissible set, hereditarily ﬁnite set, natural ordinal

In [1] the notion of Σ-reducibility on admissible sets was introduced that preserves the Σ-theory. Its main merit, as well as demerit, is the preservation of the structural properties of an admissible set such as the height of an admissible set and the structure of elements. In this paper we study some reducibility that preserves the Σ-theory rather than the particularities. It is this reducibility that will be called Σ-reducibility here. The notion of Σ-reducibility was introduced in [2]. It turned out that to study the most important properties of this reducibility we can consider only the hereditarily ﬁnite sets, i.e., the inclusion-least admissible sets. We show that for each admissible set there is an equivalent hereditarily ﬁnite set preserving a series of descriptive set-theoretic properties; in particular, the reduction principle and the existence of a universal Σ-function. As a corollary of this transformation, we give a series of lattice-theoretic properties. The main result and its corollaries were announced in [3]. However, the signature estimation is improved here. A plenary talk on the conference “Mal cev Meeting–2004” was also dedicated to these results. The lattice-theoretic and structural properties of this reducibility were studied earlier [4–7]. It turned out that the reducibility coincides on countable admissible sets with the reducibility which was introduced in [8]. It acts on the classes of arbitrary admissible sets in the same way as the reducibility suggested in [9]. We introduce the notion of jump of an admissible set which translates the descriptive set-theoretic properties in the corresponding properties lowering levels by 1. 1. Preliminaries 1.1. Computability and e-reducibility. We use some standard terminology that can be found, e.g., in [10, 11]. We just give the speciﬁc notions to be used in this paper. By we denote equality by deﬁnition. f : A → B and f : A B mean that a map f is injective and surjective respectively. By ω we denote the set of naturals. Let ·, · be a computable function yielding a one-to-one correspondence between ω × ω and ω. The author was supported by the Russian Foundation for Basic Research (Grants 06–01–04002–NNIOa and 05–01–00481), the State Maintenance Program for the Leading Scientiﬁc Schools of the Russian Federation (Grant NSh–4787.2006.1), and the Russian Science Support Foundation. Novosibirsk. Translated from Sibirski˘ı Matematicheski˘ı Zhurnal, Vol. 50, No. 2, pp. 415–429, March–April, 2009. Original article submitted September 1, 2007. 330

c 2009 Springer Science+Business Media, Inc. 0037-4466/09/5002–0330

As usual, by the join A ⊕ B we mean {2x | x ∈ A} ∪ {2x + 1 | x ∈ B}. By P(X) we denote the set of all subsets of X. We often identify functions with their graphs. Given a partial function ϕ, by δϕ, ρϕ, and Γϕ we will denote the domain, range, and graph of ϕ respectively. As usual, by enumeration reducibility (or, in short, e-reducibility) we mean the reducibility on sets of naturals which is denoted by ≤e and deﬁned as A ≤e B ⇔ ∀t (t ∈ A ⇔ ∃D ( t, D ∈ W & D ⊆ B)) for some computably enumerable set W (here D is a ﬁnite subset of naturals which can be identiﬁed with a number in the strong table). The relation ≤e is a preorder on P(ω) which naturally induces ordering on the set of e-degrees P(ω) / , where A ≡ B ⇔ A ≤ B & B ≤ A. Given A ⊆ ω, by d (A) we denote an e-degree that ≡e e e e e contains A. The collection of e-degrees forms the upper semilattice Le with bottom under the order induced by ≤e . Notice that de (A) de (B) = de (A ⊕ B) where a b is the supremum of e-degrees a and b, the bottom 0 contains exactly all computably enumerable sets. Let L = L, ≤, , 0 be an upper semilattice with bottom. A nonempty collection I ⊆ L is called an ideal of L provided that (1) a ≤ b & b ∈ I ⇒ a ∈ I; (2) a, b ∈ I ⇒ a b ∈ I. An ideal I is called principal if I is generated by some element b ∈ I; i.e., I = {c ∈ L | c ≤ b} (we will Otherwise, I is called nonprincipal. denote this ideal by b). Let L be an upper semilattice. Notice that the collection J (L) of all ideals of L forms a lattice under ⊆ and I1 ∗ I2 {x ∈ L | ∃i1 ∈ I1 , ∃i2 ∈ I2 [x ≤ i1 i2 ]}, I1 ∗ I2 I1 ∩ I2 with bottom {0} and top L. An upper semilattice L = L, ≤, , 0 with bottom is called distributive if a, b0 , b1 ∈ L have the following property: if a ≤ b0 b1 then there are c0 , c1 ∈ L such that a = c0 c1 and c0 ≤ b0 , c1 ≤ b1 . Given A ⊆ ω, put K(A) {x ∈ ω | x ∈ Φx (A)},

J(A) K(A) ⊕ (ω \ K(A)).

a

Given A ∈ a, we call de (J(A)) the e-jump of a. Given I ∈ J (Le ), put I∗ {S ⊆ ω | de (S) ∈ I}. 1.2. Elements of the theory of admissible sets. We use the standard terminology that can be found in [12, 13]. We give here just some notation and propositions from [5]. By A, B, C, . . . (possibly, with indices) we denote admissible sets with domains A, B, C, . . . (with the same indices). By an admissible set A we mean a structure of KPU whose domain is well founded under ∈. Therefore, the set of all ordinals Ord A of it is well ordered. By computably enumerable (computable) sets we mean sets that are deﬁnable by some Σ-formula (a Σ- and Π-formula simultaneously). Computably enumerable (computable) subsets are called Σ- (Δ-)subsets. For an admissible set A, the collections of all n-ary Σ- and Δ-predicates of A are denoted by Σ(An ) and Δ(An ) respectively. We omit n for n = 1. Hereditarily ﬁnite sets form an important class of admissible sets. We can deﬁne HF (M ) inductively: HF0 (M ) = M ; HFn+1 (M ) = HFn (M ) ∪ Pω (HFn (M )); HF (M ) = HFn (M ), n<ω

of predicate signature σ and where Pω (X) is the set of all ﬁnite subsets of X. If M is a structure σ ∩ ∅, ∈2 , U01 = ∅ then the structure HF(M) of signature σ ∗ = σ ∪ ∅, ∈2 , U01 can be deﬁned so that HF(M)

HF (M ) is its domain and U0 = M . It is called a hereditarily finite set over M. From now on we assume that all admissible sets are considered only in ﬁnite languages. Given an admissible set A, notice that ω ⊆ Ord A is Δ on A. In [5] all possible collections are described of the subsets of naturals realized as the collection of all Σ-subsets of ω in some admissible set. 331

Theorem 1.1. 1. For every admissible set A, the collection of all Σ-subsets of ω is represented as I∗ for some e-ideal I. 2. Given an e-ideal I, there is a structure M such that I∗ = Σ(HF(M)) ∩ P(ω). Furthermore, M can be chosen so that card(M) = card(I∗ ). By Ie (A) we denote {de (B) | B ⊆ ω, B ∈ Σ(A)}. Sometimes this ideal is called the ideal of A. A collection S ⊆ P(A) is called computable in A if S ∪ {∅} = {ΦA [a, x] | a ∈ A} for some Σ-formula Φ(x0 , x1 ). By Sω (A) we denote the class of all collections of subsets of ω computable in A. Now we give the deﬁnitions of some reducibilities on admissible sets. Let M = M, P1 , . . . , Pk be a structure and let A and B be admissible sets. (Yu. L. Ershov) We say that M = M, P1 , . . . , Pk is Σ-definable in A (M ≤Σ A) if there is a map ν : A M such that ν −1 (=), ν −1 (P1 ), . . . , ν −1 (Pk ) are Δ on A. This notion can be considered as some generalization of the notion of computable structure. (A. S. Morozov) We say that A is HYP-reducible to B (A HYP B) if there is a map ν such that A ≤Σ B via ν; there is a binary Σ-predicate R on B such that pr21 (R) = B and a, b ∈ R ⇒ ν(a) = {ν(z) | z ∈ b} for all a, b ∈ B. In fact, the reducibility is considered only on admissible sets of special kind where it suﬃces to consider a Σ-function instead of a binary Σ-predicate. This reducibility is stronger than Σ-deﬁnability; namely, A HYP B implies A ≤Σ B but not conversely [1]. In [1] this reducibility is called Σ-reducibility; however, by Σ-reducibility we mean here some reducibility that is intermediate between those above. As usual, by the ordered pair a, b we mean {{a}, {a, b}}; ∅, a1 a1 , a1 , a2 , . . . , an a1 , a2 , . . . , an if n ≥ 2; prni (a1 , a2 , . . . , an ) ai , i ≤ n. In this paper, the end of a proof is denoted by . Reasoning, omitted in a proof, can be easily restored by the reader or are analogous to the author’s arguments. 2. Admissible Sets Let A be an admissible set. We deﬁne some hierarchy of deﬁnable predicates on A as follows: • R ∈ ΣA 1 iﬀ R is a Σ-predicate on A; A • R ∈ ΣA n+1 iﬀ R = {x1 , . . . , xm | ∃yQ(y, x1 , . . . , xm )} for some m ≥ 1 and Q ∈ Πn , n ≥ 1; A ¬ A • R ∈ Πn iﬀ R ∈ Σn , n ≥ 1; A A • ΔA n Σn ∩ Πn , n ≥ 1. Theorem 2.1. Let A be an admissible set. The following relations hold on the classes of this hierarchy on A:

Moreover, all inclusions are strict. Furthermore, the following hold (m ≥ 1): A A (1) for every n ≥ 1 there is an n + 1-ary ΣA m - (Πm -)predicate universal for the class of n-ary Σm A (Πm -)predicates; A (2) for every n ≥ 1, there is no n + 1-ary ΔA m -predicate universal for the class of n-ary Δm -predicates; A ¬ (3) Σm is closed under ∧, ∨, ∃x, ∃x ∈ a but unclosed under ; ¬ (4) ΠA m is closed under ∧, ∨, ∀x, ∀x ∈ a but unclosed under ; ¬ (5) ΔA m is closed under ∧, ∨, ; A (6) R(x1 , . . . , xn ) ∈ Σm iﬀ there is Q(y, y1 , . . . , yn ) ∈ ΔA m such that R = {a1 , . . . , an | ∃y Q(y, a1 , . . . , an )}, n ≥ 1; A (7) R(x1 , . . . , xn ) ∈ ΠA m , iﬀ there is Q(y, y1 , . . . , yn ) ∈ Δm such that R = {a1 , . . . , an | ∀y Q(y, a1 , . . . , an )}, n ≥ 1. 332

We say that R ⊆ An , n ≥ 1, is definable in A if R ∈ m ΣA m . There are admissible sets for which A A A Σm , Πm , and Δm are unclosed under the restricted quantiﬁers missing in (3)–(5) of Theorem 2.1 for some m ≥ 1. The relevant examples are given below. We ﬁrst deﬁne the auxiliary hierarchy of deﬁnable predicates on A (n ≥ 1): • SnA is the least class containing ΣA n and closed under ∧, ∨, ∃x, ∃x ∈ a, ∀x ∈ a; • R ∈ PnA iﬀ ¬ R ∈ SnA ; • DnA = SnA ∩ PnA . The Σ-reﬂection principle implies A A A A Corollary 2.1. S1A = ΣA 1 , P1 = Π1 , and D1 = Δ1 for every admissible set A. A A A A Proposition 2.1. If A satisﬁes full collection then SnA = ΣA n , Pn = Πn , and Dn = Δn for all n ≥ 1. HF(M)

Corollary 2.2. If M is a structure in some ﬁnite signature then Sn HF(M) HF(M) HF(M) Πn , and Dn = Δn for all n ≥ 1.

HF(M)

= Σn

HF(M)

, Pn

=

We now deﬁne some descriptive set-theoretic properties on admissible sets. Let S be a collection of predicates of an arbitrary nature which is closed under ∩, ∪, × and contains the empty set ∅, an inclusiongreatest set U n. We say that S satisﬁes • uniformization if for every binary relation R ∈ S there is a partial function ϕ such that Γϕ ∈ S, Γϕ ⊆ R, and δϕ = pr21 (R); • reduction if for all A0 , A1 ∈ S there are disjoint sets B0 , B1 ∈ S such that Bi ⊆ Ai , i = 0, 1, and A0 ∪ A1 = B0 ∪ B1 ; • separation if for every two disjoint sets A0 , A1 ∈ S there is B ∈ S such that U n \ B ∈ S and A0 ⊆ B, A1 ⊆ U n \ B; • extension if for every unary partial function ϕ with Γϕ ∈ S there is ψ, Γψ ∈ S such that δψ = U n and Γϕ ⊆ Γψ ; • existence of a function universal for a class K of unary partial functions (graphs of all functions from K lie in S), if there is a binary partial function ψ, Γψ ∈ S, such that K = {λy. ψ(a, y) | a ∈ U n}. • An admissible set A is called Σn -listed (via ω), if there is a ΣA n -function f from ω onto A. • An admissible set A is called n-quasiprojectible (into ω), if there is a ΣA n -function f from some R ⊆ ω onto A. These properties will be called basic (we consider here only two classes as K: of all partial functions and all partial {0, 1}-valued functions). Let A be an admissible set and let P be one of the basic properties (except the last two). We say that A satisﬁes n-P if ΣA n satisﬁes P , n ≥ 1. If P is quasiprojectibility then the meaning of n-P is clear. In case when P is enumerability, n-P means that A is Σn -listed. The Σ1 -listed admissible sets will be recursively listed. These admissible sets are actively studied in [13]. In case n = 1, we say that A satisﬁes P instead of 1-P . Given n the relations between the basic properties of kind n-P are the same as for n = 1. If A is projectible into ω then A is quasiprojectible into ω [13]. Proposition 2.2. Let A be an admissible set and let n ≥ 1. The following hold: (1) if A is Σ1 -listed then A is a hereditarily ﬁnite set; (2) if A is Σn -listed then A is Σk -listed for every k ≥ n; (3) if A is n-quasiprojectible then A is Σn+1 -listed; A (4) if A is Σn -listed and Ord(A) > ω then m ΣA m = Sn . Proof. 1. Suppose to contrary that there is a Σ1 -listed admissible set but not hereditarily ﬁnite set. Then there are an inﬁnite element a ∈ A and Σ-functions f0 : ω A, f1 : a ω. Note that ω ∈ A by Σ-replacement to f1 [13, Theorem 4.6]. Again apply Σ-replacement to f0 . We have A ∈ A, which is a contradiction. 333

4. This follows from the fact that each formula ∀xϕ is equivalent on A to ∀k ∈ ω∃x((f (k) = x) ∧ ϕ), where f is a ΣA n -function enumerating A via ω and k is not free in ϕ. It follows from Proposition 5.2 and [6, Theorem 3.1] that generally n-P does not imply (n + 1)-P where P is separation or extension. Proposition 2.3. If n ≥ 1 and A is an n-quasiprojectible admissible set then A does not satisfy n-extension. Proof. Let A be n-quasiprojectible. If A is Σn -listed then there is a universal ΣA n -function f (x, y), -functions where and so sg(f (x, x)) has no extension in the class of ΣA n ∅, if x = ∅, sg(x) = {∅}, if x = ∅. Now, let A be not Σn -listed. There is a ΣA n -function f : R A for some R ⊂ ω. It has no extension A in the class of Σn -functions; otherwise, h ω would be the ΣA n -function enumerating A where h is an extension of f . In [13] a series A of admissible sets projectible into ω ∈ A is given. HYP(N) is such an admissible set where N is the standard model of arithmetic. These admissible sets satisfy (4) of Proposition 2.2. 3. Σ-Reducibility: Deﬁnition and Basic Properties Let A and B be admissible sets. Definition 3.1. We say that A is Σ-reducible to B (A Σ B) if ν −1 (Σ(A2 )) ⊆ Σ(B2 ) for some ν : B A. In this event we say also that A is Σ-reducible to B via ν (ν : A Σ B). Corollary 3.1. Σ is reﬂexive and transitive. We say that A and B are Σ-equivalent (A ≡Σ B) if A Σ B and B Σ A. [1, Lemma 1] implies Corollary 3.2. If A HYP B then A Σ B. The converse of Corollary 3.2 does not hold, which follows from [1, Proposition 1] and Theorem 3.1. Corollary 3.3. If A Σ B then A ≤Σ B. The converse of Corollary 3.3 does not hold either; e.g., HYP(N) ≤Σ HF(HYP(N)) whereas HYP(N) Σ HF(HYP(N)) [7]. The classical computability is the least element under the above-introduced level of complexity. Proposition 3.1. HF(∅) Σ A for every admissible set A. Hence, the class of Σ-degrees has bottom under Σ-reducibility. Proposition 3.2. ν : A Σ B iﬀ ν −1 (Σ(An )) ⊆ Σ(Bn ) for all n < ω. Proof. (⇒) is clear for n = 2; for n = 1 it follows from the fact that Σ-predicates are closed under projections and cartesian products. Let n > 2 and C ∈ Σ(An ). Denote by C (∈ Σ(A2 )) the predicate (x), 1 ≤ i < n, is a Σ{x1 , . . . , xn−1 , xn | C(x1 , . . . , xn−1 , xn )}. Since the ith projection prn−1 i n−1 n−1 n−1 , it follows that ν −1 (C) = y , . . . , y , y | ∃y (ν(y1 )) = function on A deﬁned on A 1 1 n−1 2 i=1 pri

n−1 ν(yi ) ∧ (ν(y1 ) ∈ A ) ∧ C (ν(y1 ), ν(y2 )) is a Σ-predicate on B. By induction on the quantiﬁer complexity, we infer the following

B −1 ΠA ⊆ ΠB , and ν −1 (ΔA ) ⊆ ΔB with arity Proposition 3.3. ν : A Σ B iﬀ ν −1 ΣA m ⊆ Σm , ν m m m m preserving for all m ≥ 1. 334

Lemma 3.1. If ν : A Σ B then R0 = {x, n | n ∈ ω ⊆ Ord(B), ν(x) = n} is Δ on B. Proof. This predicate can be deﬁned by Σ-recursion: x, 0 ∈ R0 ⇔ ν(x) = ∅ ⇔ ¬ (ν(x) = ∅); x, n + 1 ∈ R0 ⇔ ∃x (x , n ∈ R0 ∧ ν(x ) + 1 = ν(x)); x, n + 1 ∈ R0 ⇔ ¬ Nat(ν(x)) ∨ (ν(x) = ∅) ∨ ∃x (x , n ∈ R0 ∧ (ν(x ) + 1 = ν(x))); where Nat(a) means “a is a natural ordinal.” Corollary 3.4. If ν : A Σ B then R1 = {x, n | n ∈ ω ⊆ Ord(B), ν(x) ∈ An } is Δ on B. Proposition 3.4. If A Σ B then Sω (A) ⊆ Sω (B). In particular, Ie (A) ≤ Ie (B). Proof. Let ∅ = S ⊆ P(ω) be computable in A and let Q ∈ Σ(A2 ) be such that S = {{n | Q(a, n)} | a ∈ A}. Then ν −1 (Q) ∈ Σ(B2 ) where ν : A Σ B. It is easy to verify that S = {{n | ∃y(y, n ∈ R0 ∧ ν(b), ν(y) ∈ Q)} | b ∈ B}, where R0 is Δ on B by Lemma 3.1.

The following assumption is immediate from ﬂattening [12]. Furthermore, it ensues from the proof of Proposition 1.2 in [4]. Proposition 3.5. Let M be a structure in some ﬁnite signature and let A be an admissible set. M ≤Σ A iﬀ HF(M) Σ A. Theorem 3.1. Let A be an admissible set. Then there is an oriented graph MA without loops, dom(MA ) = A, such that the following hold (n ≥ 1): (1) A ≡Σ HF(MA ); (2) A satisﬁes n-P iﬀ HF(MA ) also satisﬁes n-P where P is a basic property. Proof. Let A be an admissible set and let U be a Σ-predicate on A universal for the class of all unary Σ-predicates. By Σ-reﬂection, there is a ternary Δ0 -predicate U such that A |= U (x, y) ≡ ∃uU (x, u, y). By “Pair” and “Triple” we denote the collections of “ﬁnite” functions f on A with δf = 2 and = 3 respectively. We deﬁne V as follows: ⎧ Pair(b), if a = 0, ⎪ ⎪ ⎪ ⎪ Triple(b) ∧ (b(0) = 0), if a = 1, ⎪ ⎪ ⎪ ⎪ ⎪ Triple(b) ∧ (b(0) = 1), if a = 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Triple(b) ∧ (b(1) = a(0)) ∧(b(2) = a(1)), if Pair(a), V (a, b) ⎪ ⎪ ⎪ a(1) = b, if Triple(a) ∧ (a(0) = 0), ⎪ ⎪ ⎪ ⎪ ⎪ a(2) = b, if Triple(a) ∧ (a(0) = 1), ⎪ ⎪ ⎪ ⎪ ⎪ U (a(1), a(2), b), if Triple(a) ∧ Pair(a(0)) ⎪ ⎪ ⎩ ∧(a(0)(0) = b). Now, let MA be A, V . It is evident that MA is an oriented graph without loops. We prove that the claims are true for this structure. 1. It is easy that V is Δ on A, and so MA ≤Σ A. By Proposition 3.5, HF(MA ) Σ A. To verify A Σ HF(MA ), it suﬃces to show that U will be Σ on HF(MA ) and Σ-functions a(0), a(1) on A, determined on Pair, are also Σ-functions on HF(MA ): a(0) = b ⇔ (V (0, a) ∧ ∃x(V (a, x) ∧ (V (1, x) ∧ V (x, b)))), a(1) = b ⇔ (V (0, a) ∧ ∃x(V (a, x) ∧ (V (2, x) ∧ V (x, b)))), U (x, y) ⇔ ∃u∃z(V (0, z) ∧ ((z(0) = x) ∧ ((z(1) = u) ∧∃a(V (z, a) ∧ (¬ V (1, a) ∧ (¬ V (2, a) ∧ V (a, y))))))). 335

2. Let P be reduction, uniformization, separation, or extension. If HF(MA ) satisﬁes n-P , n ≥ 1, then so does A. Conversely, we only consider separation. We ﬁrst give an auxiliary construction. Lemma 3.2. There is an embedding ı : HF (A) → A that is Σ on HF(MA ) and ı(HF (A)) ∈ Δ(HF(MA )). Proof of Lemma 3.2. It follows from Theorem 1 in [14] that there is a partial Σ-function Term : Ord(HF(MA ))×A<ω HF (A) such that the domain of Term is Δ and, for every a ∈ HF (A), Term−1 (a) is ﬁnite. Furthermore, Term acts one-to-one on the ﬁrst coordinate. If Term(n, a) = x then sp(a) = sp(x) and the coordinates of a are distinct. Also, there is a strongly computable sequence of ﬁnite groups {Sn }n<ω such that Term(n, a) = Term(n, b), iﬀ there exists a permutation π ∈ Sn , for which π(a) = b. We now deﬁne ı as follows: let x, n, a ∈ HF (A) be such that x = Term(n, a). Then ı(x) = n, {π(a) | π ∈ Sn } ∈ A. It is easy to verify that ı is a Σ-function on HF(MA ) with the desired properties. We continue the proof of Theorem 3.1. Let A satisfy n-separation. We show that so does HF(MA ). HF(MA ) Take disjoint Σn -subsets A0 and A1 . Given ı(A0 ) and ı(A1 ) there exists a ΔA n -subset B such that HF(MA ) −1 −1 ı(A0 ) ⊆ B ⊆ ı(A1 ). It remains to check that ı (B) is Δn and A0 ⊆ ı (B) ⊆ A1 . HF(MA ) -function. Then Let ϕ(x, y) be a universal Σn ϕ(ı−1 (x), y), if y ∈ A, ϕ(ı−1 (x), y)↓ ∈ A, ψ(x, y) ↑ otherwise A −1 is a universal ΣA n -function. Now, if f (x, y) is a universal Σn -function then g(x, y) ı (f (x, ı(y))) is HF(MA ) a universal Σn -function. If A is Σn -listed then there is a ΣA n -function f : ω A. By Lemmas 3.2 and 3.1, it is easy to HF(MA ) construct a Σn -function that enumerates HF (A).

Remark 3.1. MA from the proof of Theorem 3.1 is independent of the choice of a universal predicate U . Namely, if U1 and U2 are Σ-predicates universal for the collection of all binary Σ-predicates then id : HF(A, V1 ) ≡Σ HF(A, V2 ) where Vi is constructed from Ui , as in the proof of Theorem 3.1, i = 0, 1. Notice that it is impossible to improve the signature estimation for MA in Theorem 3.1, since all structures consisting of ﬁnitely many subsets are locally constructivizable. Theorem 3.2. Let A and B be admissible sets. Then A Σ B iﬀ M ≤Σ A implies M ≤Σ B for every structure M in some ﬁnite signature. Proof. (⇒): If A Σ B and M ≤Σ A then HF(M) Σ A by Proposition 3.5; and so HF(M) Σ B by transitivity. By Proposition 3.5 again, M ≤Σ B. (⇐): MA ≤Σ A, by Theorem 3.1 and Proposition 3.5. Hence, MA ≤Σ B. Applying Theorem 3.1, Proposition 3.5, and the transitivity of Σ-reducibility, we see that A Σ B. 4. Σ-Jump: Deﬁnition and Basic Properties By J (A) we denote HF(A, U ) where U ⊆ A3 is Σ on A universal for the class of all binary Σpredicates. J (A) is called the Σ-jump of A. Remark 4.1. The Σ-jump operation is correctly deﬁned; namely, if U1 and U2 are ternary universal Σ-predicates on A then HF(A, U1 ) ≡Σ HF(A, U2 ) via id. Lemma 4.1. Let A be an admissible set and let MA be the model constructed in the proof of HF(M ) J (A) Theorem 3.1. Then Σn+1 A = Σn for every n < ω. Proof. Let MA = A, V and J (A) = HF(A, U ) be the structures from the lemma. Applying Remarks 3.1 and 4.1, we assume that U = {a, b(0), b(1) | A |= ∃uV ({0, {0, b, 1, 0}, 1, a, 2, u}, b) ∧ Pair(b)}. 336

n = 1. Let B ⊆ Ak be deﬁnable by some Σ-formula Φ(x1 , x2 , . . . , xk ) in J (A), k ≥ 1. We can assume that all negations appear just before atomic subformulas and all implications are absent. By full collection for HF(MA ), [Φ]U Ψ is equivalent to some Σ2 -formula in HF(MA ) where Ψ is a Σ-formula in signature {V } for which J (A) |= U (x, y, z) ⇔ HF(MA ) |= Ψ(x, y, z). Conversely, let C be Σ2 on HF(MA ). Then ı(C) ⊆ A is also Σ2 on HF(MA ) (where ı is deﬁned in Lemma 3.2). Hence, by Proposition 3.3 there is a Σ-formula Θ(x, y) such that ı(C) = {a | A |= ∃u¬ Θ(u, a)}. Since U is universal, there is a0 ∈ A such that A |= Θ(u, a) ≡ U (a0 , u, a). Thus ı(C) ∈ Σ(J (A)) and so C = ı−1 (ı(C)) ∈ Σ(J (A)). Since the projections are Σ-functions on HF(MA ) and J (A), the lemma holds for the remaining Σ2 -predicates. It suﬃces to apply induction. By Lemma 4.1 and the transitivity property of Σ-reducibility, we have Theorem 4.1. Let A and B be admissible sets. Then

(1) B Σ J (A) iﬀ there is ν0 : A B such that ν0−1 ΣB1 ⊆ ΣA 2 with arity preserving; A

−1 (2) J (A) Σ B iﬀ there is ν1 : B A such that ν1 Σ2 ⊆ ΣB1 with arity preserving. By Lemma 4.1 and Theorem 3.1, we have the following Theorem 4.2. Let A be an admissible set. A satisﬁes (n + 1)-P iﬀ J (A) satisﬁes n-P , where P is a basic property, n ≥ 1. As in the classical case, the Σ-jump operation satisﬁes the following: (1) A Σ J (A); (2) A Σ B ⇒ J (A) Σ J (B). We deﬁne J n (A) by induction on n < ω as follows: J 0 (A) A, J n+1 (A) J (J n (A)). We say that M = M, Q1 , . . . , Qs , s ∈ ω, is definable in A if there is ν : A M such that ν −1 (=), ν −1 (Q1 ), . . . , ν −1 (Qs ) are deﬁnable on A. This notion was introduced by Yu. L. Ershov. The deﬁnability is a preorder on the class of admissible sets. Notice that this relation is weaker than Σ-deﬁnability. We introduce more notions of deﬁnability in admissible sets. Let m ≥ 1. We say that M = M, Q1 , . . . , Qs , s ∈ ω, is Σm -definable in A if there exists ν : A M such that ν −1 (=), ν −1 (Q1 ), . . . , ν −1 (Qs ) are in ΔA m . Notice that Σ1 -deﬁnability coincides with Σ-deﬁnability. Applying Lemma 4.1 several times we have the following Theorem 4.3. Let A be an admissible set, let m ∈ ω, and let M be a structure in some ﬁnite signature. M is Σm+1 -deﬁnable in A iﬀ M is Σ-deﬁnable in J m (A). Theorem 4.4. Let A be an admissible set and let M be a structure in some ﬁnite signature. M is deﬁnable in A iﬀ M is Σ-deﬁnable in J n (A) for some n < ω. 5. Structures Having e-Degree Let A ⊆ ω. We deﬁne NA in signature {F 2 , 01 , s2 } with domain NA as follows: NA ω {zn | n ∈ A}, where zn = zn whenever n = n ; 0NA {0} ⊆ ω; sNA {n, n + 1 | n ∈ ω}; F NA {x, y | x ∈ A, y = zx }. These structures have e-degrees. The notion of structure having e-degree can be found in [9]. In fact, they are studied earlier [15]. The deﬁnition, based on representations on naturals, is not given here. Notice only that a countable structure M (in some ﬁnite language) has e-degree iﬀ Ie (HF(M)) is principal and HF(M) has the minimal property for Ie (HF(M)) (it means that for every admissible set A, if Ie (A) = Ie (HF(M)) then HF(M) Σ A). The properties of these structures are actively studied in [6]. Proposition 5.1. Let A, B ⊆ ω. HF(NA ) Σ HF(NB ) iﬀ A ≤e B. The proof is immediate from [6, Theorem 3.1] and Proposition 3.4. The following theorem gives a description of the admissible sets that can be eﬀectively enumerated via naturals. HF(N), HYP(N), and Lα are examples of these structures where N is the standard model of arithmetic and α is an admissible ordinal projectible into ω [13]. 337

Theorem 5.1. Let A be a quasiprojectible admissible set. Then (1) Ie (A) is principal. (2) Let C ⊆ ω be such that Ie (A) = d e (C). Then A ≡Σ HF(NC ). (3) A does not satisfy the extension principle. Proof. Let h : R A be a Σ-function from the deﬁnition of quasiprojectible admissible set where R ⊆ ω. 1. Let Uω ⊆ A × ω be Σ on A universal for the collection of all Σ-subsets of ω. Then de ({ m, n | m ∈ R, h(m), n ∈ Uω }) ∈ Ie (A) and so Ie (A) is principal. 2. HF(NC ) Σ A follows from the proof of Theorem 3.1 [6] (HF(NC ) has the minimal property). Now we prove that A Σ HF(NC ). Let C R ⊕ { m, n | m ∈ Φn (C)}. It is evident that C ≡e C and A Σ HF(NC ) via h(n), if x = 2 · n, z2·n for some n ∈ R, ν(x) ∅ otherwise. By the transitivity property of Σ-reducibility and Proposition 5.1, we have the desired property. Proposition 5.2. HF(NJ(C) ) ≡Σ J (HF(NC )) for every C ⊆ ω. Proof. Let C ⊆ ω. Then J(C) ∈ Σ(J (HF(NC ))) and so HF(NJ(C) ) Σ J (HF(NC )). Hence, J (HF(NC )) is Σ-listed. So, to prove J (HF(NC )) Σ HF(NJ(C) ), it suﬃces to show that (J(C)) Ie (J (HF(NC ))) ≤ de HF(N )

C ∩ P(ω). There is a Σ-formula Φ(x0 , x1 ) such that n ∈ A ⇔ by Theorem 5.1. Take A ∈ Σ2 ∃x0 ¬ Φ(x0 , n) ⇔ ∃m(m ∈ R ∧ ¬ Φ(h(m), n)) where h : R HF (NC ) is a Σ-function on HF(NC ), R ⊆ ω. We can assume that it has no parameter since parameters can be eliminated in HF(NC ). We get a formula equivalent to some Σ-formula in HF(N, K(C)) where N is the standard model of arithmetic.

6. Σ-Reducibility: Algebraic Properties In this section we consider some algebraic properties of admissible sets under Σ-reducibility. Corollary 6.1. For all admissible structures A0 and A1 , there is an admissible set A0 A1 such that A0 Σ A0 A1 , A1 Σ A0 A1 and A0 Σ B, A1 Σ B ⇒ A0 A1 Σ B. Moreover, Ie (A B) = Ie (A) Ie (B). Proof. Let HF(MA0 ) and HF(MA1 ) be the hereditarily ﬁnite superstructures as in Theorem 3.1. We deﬁne the hereditarily ﬁnite set A0 A1 over M in signature {P 1 , Q2 } as follows: M MA0 MA1 ; P M MA0 ; QM QMA0 ∪ QMA1 . It is easy to verify that A0 A1 has the desired properties. The last condition is immediate from [5, Proposition 3.1]. The class of admissible sets under Σ containing A is denoted by [A]Σ . By we denote the order that is induced by Σ . We consider the following orderings for an inﬁnite cardinal α and an e-ideal I, card(I) ≤ α: • Lα {[A]Σ | A is admissible and card(A) = α}, ; • L≤α {[A]Σ | A is admissible and card(A) ≤ α}, ; • Lα,I {[A]Σ | A is admissible and card(A) = α, Ie (A) = I}, ; • L≤α,I {[A]Σ | A is admissible and card(A) ≤ α, Ie (A) = I}, . We give a series of basic properties of these orderings: (1) they yield upper semilattices; (2) Lα and L≤α are closed under the Σ-jump operation for every inﬁnite α; however, Lα,I and L≤α,I are closed under the Σ-jump operation just for I = Le and α ≥ 2ω (as follows from [16]); (3) Lα , L≤α , and Lα,I have bottom for all inﬁnite α and e-ideal I, with card(I) ≤ α (as follows from Theorem 3.9 in [5]); 338

(4) L≤α,a has bottom for all inﬁnite α and e-degree a (as follows from Theorem 3.1 in [6]); (5) Let α be a cardinal and let I be a nonprincipal e-ideal I such that card(I) < α. Then L≤α,I has card({card(β) | card(I) ≤ β ≤ α}) many minimal elements and so L≤α,I is not a lattice. (6) card(Lα ) = 2α (as follows from Theorem 3.1 in [6]). All possible identical embeddings are not given here. Now, we give a transformation of an admissible set to another one of greater cardinality which preserves a series of properties, in particular, the computable invariants such as the ideal of the admissible set and the class of computable collections of P(ω). Let A be an admissible set of cardinality β and let S be a set with no additional structure of cardinality α ≥ β. By Aα we denote the hereditarily ﬁnite superstructure over M = MA S, QMA , S where MA = MA , QMA is a structure from Theorem 3.1. Theorem 6.1. Let A and B be admissible structures of cardinality ≤ α and let card(A) ≤ card(B). Then (1) A Σ B iﬀ Aα Σ Bα ; (2) (A B)α = Aα Bα ; (3) J (Aα ) ≡Σ J (A)α ; (4) I(A) = I(Aα ). Proof. 1. Let Aα be a hereditarily ﬁnite set over A. By Proposition 3.5, A ≤Σ Bα . Take some X ⊆ ν −1 (MA ) for which card(X) = card(MA ) and ν(X) = MA where A ≤Σ Bα via ν. Now, let Y be a set of cardinality card(B) which contains X and also some subset of S of cardinality card(B) such that all parameters of the formulas from the deﬁnition of A ≤Σ Bα are included owenheim–Skolem in it. By the L¨ Theorem, there is B0 Bα of cardinality card(B) such that MB ∪ {sp(y) | y ∈ Y } ⊆ B0 . It is evident that B0 ≡Σ B. Deﬁne A0 in B0 by the same formulas as A in B. Then A Σ HF(A0 ), A0 ≤Σ B0 and so A Σ B. 3. Evidently, J (A)α Σ J (Aα ). Conversely, it is easy to construct ν : MA S0 , QMA , S0 ≤Σ A where S0 ⊆ S has cardinality ω and the elements from S0 are enumerated via ν by naturals. Also, we have HF(MA S0 , QMA , S0 ) HF(MA S, QMA , S). Moreover, the collections of the types realized in these structures coincide. The remaining claims follow from the fact that there is ν ∗ : J (HF(MA S0 , QMA , S0 )) Σ J (A) “extending” ν (in other words, ν ∗ is constructed from ν by the approach proposed, e.g., in [4]). As corollaries of this theorem, we give a series of embeddings of semilattices for every e-ideal I and all cardinals β ≤ α: (1) Lβ , J → Lα , J ; (2) L≤β , J → Lα , J ; (3) Lβ,I → Lα,I ; (4) L≤β,I → Lα,I ; (5) in particular, Lβ is not distributive, because, by Proposition 5.1, the upper semilattice Le of enumeration degrees is embeddable into it but Le is not distributive; (6) Le , → Lω , J by Propositions 5.1 and 5.2. 7. Open Problems 1. Is the collection of all Σ-degrees of countable admissible sets a lattice? 2. Describe the lattice-theoretic properties of the Σ-degrees of locally constructivizable hereditarily ﬁnite structures. 3. Is there a computable admissible set not Σ-equivalent to HF(∅)? 4. Does the Σ-jump operation have ﬁxed points? 5. Describe the image of the Σ-jump operation. 6. Is there an embedding ı of J (Le ) to L2ω , considered as an upper semilattice, such that Ie (ı(I)) = I, for every e-ideal I? 339

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