The groups \mathbb{Z} and \mathbb{Q} with predicates for being square-free

The additive groups of and with predicates for being square-free

Neer Bhardwaj, Minh Chieu Tran Department of Mathematics, University of Illinois at Urbana- Champaign, Urbana, IL 61801, U.S.A nbhard4@illinois.edu, mctran2@illinois.edu
July 14, 2019
Abstract.

We consider the four structures , , , and where is the additive group of integers, is the set of such that for every prime and corresponding -adic valuation , and are defined likewise for rational numbers, and denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

2010 Mathematics Subject Classification:
Primary 03C65; Secondary 03B25, 03C10, 03C64

1. Introduction

In [KS16], Kaplan and Shelah showed under the assumption of Dickson’s conjecture that if is the additive group of integers implicitly assumed to contain the element as a distinguished constant and the map as a distinguished function, and if Pr is the set of such that either or is prime, then the theory of is model complete, decidable, and super-simple of U-rank . From our current point of view, the above result can be seen as an example of a more general phenomenon where we can often capture aspects of randomness inside a structure using first-order logic and deduce in consequence several model-theoretic properties of that structure. In , the conjectural randomness is that of the set of primes with respect to addition. Dickson’s conjecture is useful here as it reflects this randomness in a fashion which can be made first-order. The second author’s work in [Tra17] provides another example with similar themes.

Our viewpoint in particular predicts that there are analogues of Kaplan and Shelah’s results with Pr replaced by other random subsets of . We confirm the above prediction in this paper without the assumption of any conjecture when Pr is replaced with the set

where is the -adic valuation associated to the prime . We have that is a structure in the language of additive groups augmented by a constant symbol for and a function symbol for . Then is a structure in the language extending by a unary predicate symbol for . We will introduce a first-order notion of genericity which encapsulates the randomness in the interaction between and the additive structure on . Using an approach with the same underlying principle as that in [KS16], we obtain:

Theorem 1.

The -theory of is model complete, decidable, super simple of -rank , and is -independent for all .

From the same notion of genericity, we deduce consequences in the opposite direction for the structure in the language extending by a binary predicate symbol for the natural ordering :

Theorem 2.

The -theory of is bi-interpretable with the theory of .

The above is an analogue of a result in [BJW93] for the structure where is the set of primes, and essentially the same proof works. Theorem 2 is not completely unexpected. Indeed, it is proven in [DG17] that there is no strong expansion of the theory of Presburger arithmetic. This lends support to the heuristic that adding a random predicate to Presburger arithmetic results in defining multiplication.

From the above picture, it is also natural to consider and where is the additive group of rational numbers, also implicitly assumed to contain as a distinguished constant and as a distinguished function, is the set , and the relation on is the natural ordering. Then can be construed as an -structure and can be construed as an -structure. (We do not study because every integer is a sum of two elements in , and so we can define in the set .) Through defining other notions of genericity for these two structures, we get:

Theorem 3.

The -theory of is model complete, decidable, simple but not super-simple, and is -independent for all .

From above, is “less tame” than . The reader might therefore expect that is wild. However, this is not the case:

Theorem 4.

The -theory is model complete, decidable, has but is not strong, and is -independent for all .

The paper is arranged as follows. In section 2, we define the appropriate notions of genericity for the structures under consideration. The model completeness and decidability results are proven in section 3 and the combinatorial tameness results are proven in section 4.

Notation and conventions

Let and range over the set of integers and let , , and range over the set of natural numbers (which include zero). We let range over the set of prime numbers, and denote by the -adic valuation on . Let be a single variable, a tuple of variables of unspecified length, the tuple of variables, and the tuple of variables. For an arbitrary language , let denote the set of first-order -formulas where the only free variables are among the components of . Suppose is an -structure and is a subset of . We let denote the language extending by adding constant symbols for elements of . By writing we implicitly assume for some and . For an -tuple of elements from a certain set, we let denote the -th component of for . For an arbitrary -structure such that is an abelian group and , we define in the obvious way and write for .

2. Genericity of the examples

We study the structure indirectly by looking at its definable expansion to a richer language. For given and , set

Let . The definition for is not too useful as in this case. However, we still keep this for the sake of uniformity as we treat later. For , set

In particular, . Let . We have that is a structure in the language extending by families of unary predicate symbols for and . It is easy to see that a subset of is definable in if and only if it is definable in .

Let be an -structure. Then is a family indexed by pairs , and is a family indexed by . For , , and , define to be the member of with index and to be the member of the family with index . In particular, we have

Clearly, this generalizes the previous definition for .

We isolate the basic first-order properties of . Let be a set of -sentences such that an -structure is a model of if and only if satisfies the following properties:

  1. is elementarily equivalent to ;

  2. for , and for ;

  3. and are in ;

  4. for any given , we have that if and only if and ;

  5. for all .

It is well-known that is decidable. Hence, we can arrange that is recursive. Clearly, is a model of . Several properties which hold in are easily seen to also hold in an arbitrary model of :

Lemma 1.

Let be a model of . Then we have the following:

  1. is elementarily equivalent to ;

  2. for all , , , and , we have that

  3. for all , , and , we have that if and only if

  4. if is in for some , then ;

  5. for all and , if and only if we have

  6. for all and , if and only if . ∎

We next consider the structures and . For given , , and , in the same fashion as above, we set

and let

Then is a structure in the language . Clearly, every subset of definable in is also definable in . A similar statement holds for and . We will show that the reverse implications are also true.

Lemma 2.

Every integer is a sum of two elements from .

Proof.

It is well-known that any is a sum of two square-free natural numbers; see [Rog64] for instance. The statement of the lemma follows. ∎

Lemma 3.

For all and , is existentially -definable in .

Proof.

As for all and , it suffices to show the statement for . Fix a prime . We have that

Using Lemma 2, for all , we have that if and only if there are such that

Hence, the set is existentially definable in . The desired conclusion follows. ∎

It is also easy to see that for all , for all , and so is existentially -definable in . Combining with Lemma 3, we get:

Proposition 4.

Every subset of definable in is definable in . The corresponding statement for and also holds.

In view of the first part of Proposition 4, we can analyze via in the same way we analyze via . Let be a set of -sentences such that an -structure is a model of if and only if satisfies the following properties:

  1. is elementarily equivalent to ;

  2. for any given , the existential formula obtained in the proof of Lemma 2.3 defines the subgroup of ;

  3. for any given , if and if ;

  4. is isomorphic as a group to ;

  5. ;

  6. for any given , we have that if and only if and .

  7. for

It is well-known that is decidable. Hence, we can arrange that is recursive. Obviously, is a model of . Several properties which hold in are easily seen to also hold in an arbitrary model of :

Lemma 5.

Let be a model of . Then we have the following:

  1. for all and with ,

  2. for all , , , , and , we have that

    where is the obvious element in and in ;

  3. the replica of (3-6) of Lemma 1 holds.

As the reader may expect by now, we will study via . Let be . Then can be construed as an -structure in the obvious way. Let be a set of -sentences such that an -structure is a model of if and only if satisfies the following properties:

  1. is elementarily equivalent to ;

  2. is a model of .

As is decidable, we can arrange that is recursive.

Returning to the theory , we see that it does not fully capture all the first-order properties of . For instance, it follows from Lemma 11 below that for all , there is such that

while the interested reader can construct models of where the corresponding statement is not true. Likewise, the theories and do not fully capture all the first-order properties of and .

To give a precise formulation of the missing first-order properties of , , and , we need more terminologies. Let be an -term (or equivalently an -term) with variables in . If is either an -structure or an -structure, and , define to be the -linear combination of the components of given by . Define in the obvious way the formulas

A boolean combination of formulas having the form where we allow to vary is called an equational condition in . Similarly, a boolean combination of formulas having the form where is allowed to vary is called an order-condition in . For any given , define to be the obvious formula in which defines in an arbitrary -structure the set

Define the quantifier-free formulas , , and in for , , and for likewise. For each prime , a boolean combination of formulas of the form where and are allowed to vary is called a -condition in . We call a -condition as in the previous statement trivial if the boolean combination is the empty conjunction.

A parameter choice in is a triple such that is in , is in , and is a family of -formulas such that is a -condition for all and is trivial for all but finitely many . We call a special formula if has the form

where and are taken from a parameter choice in . Every special formula in corresponds to a unique parameter choice in and vice versa.

Let be the special formula corresponding to a parameter choice with . Let be the formula

We call the -condition associated to . It is easy to see that is a logical consequence of .

Suppose is a special formula, is its associated -condition for each prime , is an -structure, and are such that the components of and are pairwise distinct. We call the quantifier-free formula a -system. The -systems are general enough to represent quantifier free formulas with parameters in and special enough that in the structures of interest we have a “local to global” phenomenon.

Suppose and are -structures such that the former is an -substructure of the latter. Let be a -system. An element such that is called a solution of in . We say that is satisfiable in if it has a solution in and infinitely satisfiable in if it has infinitely many solutions in . For a given , we say that is -satisfiable in if there is such that . A -system is locally satisfiable in if it is -satisfiable in for all .

Suppose and are -structures such that the former is an -substructure of the latter. All the definitions in the previous paragraph have obvious adaptations to this new setting as and are -structures. For and in such that , define

A -system is satisfiable in every -interval if it has a solution in the interval for all and in such that . The following observation is immediate:

Lemma 6.

Suppose is a model of either or . Then every -system which is satisfiable in is also locally satisfiable in .

It turns out that the converse and more are also true for the structures of interest. We say that a model of either or is generic if every locally satisfiable -system is infinitely satisfiable in . A model is generic if every locally satisfiable -system is satisfiable in every -interval. We will later show that , , and are generic.

Before that we will show that the above notions of genericity are first-order. Let be the special formula corresponding to a parameter choice with . A boundary of is a number such that and is trivial for all .

Lemma 7.

Let be a special formula, a boundary of , and a model of either or . Then every -system is -satisfiable for .

Proof.

Let be the special formula corresponding to a parameter choice , and , as in the statement of the lemma. Suppose is a -system, , and is the -condition associated to . Then

We will show the stronger statement that there is satisfying the latter. As a consequence of this strengthening, we can assume that for . In light of Lemma 1 (1) and Lemma 5 (1), we have that

It is easy to see that is invertible and . Choose in such that the images of in are not . We check that is as desired. ∎

Corollary 8.

There is an -theory such that the models of are the generic models of . Similarly, there is an -theory and an -theory satisfying the corresponding condition for and .

In the rest of the paper, we fix , , and to be as in the previous lemma. We can moreover arrange them to be recursive. In the remaining part of this section, we will show that , and are models of , , and respectively. The proof that the latter are in fact the full axiomatizations of the theories of the former needs to wait until next section.

Suppose and is a boolean combination of atomic formulas of the form or where is an -term with variables in . Define to be the formula obtained by replacing and in with and for every choice of , , and -term . By construction, across -structures,

Moreover, if is a -condition, then is a -condition and if is the special formula corresponding to a parameter choice with , then is the special formula corresponding to the parameter choice with . It is easy to see from here that:

Lemma 9.

Any boundary of a special formula is also a boundary of .

Let be a special formula, a model of either or , and a -system. Then is also a -system which we refer to as the -conjugate of . This has the property that if and only if for all .

For and in , we write if and have the same remainder when divided by . We need the following version of Chinese remainder theorem:

Lemma 10.

Suppose is in , where is a -condition for all , and is such that defines a nonempty set in for all . Then we can find such that for all with , for some we have that

Proof.

Let , , and be as stated. Fix such that . For each , the -condition is a boolean combination of atomic formulas of the form where is an -term with variables in . For , let be the largest value of occurring in an atomic formula in . As , it is easy to see that is independent of the choice of . Set

Since defines a nonempty set in , so does . Obtain such that holds. By the Chinese remainder theorem, we get in such that

Suppose is such that . By construction, if and is an atomic formula in , then if and only if . It follows that holds for all . The desired conclusion follows. ∎

Towards showing that the structures of interest are generic, the key number-theoretic ingredient we need is the following result:

Lemma 11.

Let be a special formula and a -system which is locally satisfiable in . For , and with , set

Then there exists , and such that for all with and with , we have that

Proof.

Throughout this proof, let , , and be as in the statement of the lemma. We first make a number of observations. Suppose corresponds to the parameter choice and has a boundary , and is the -condition associated to . Then corresponds to the parameter choice , and is also a boundary of by Corollary 9. Moreover is the -condition associated to . Using Lemma 10, there is such that for with , for some we have that

We emphasize that here is independent of the choice of for all with .

We introduce a variant of which is needed in our estimation of . Until the end of the proof, set . Fix primes such that , for all , for all and

For and , define to be the set of such that and

It is not hard to see that