The modal logic of abelian groups

We prove that the modal logic of abelian groups with the accessibility relation of being isomorphic to a subgroup is S4.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {S4.2}$$\end{document}.


Introduction
Let L be a first-order language, A be a collection of L-sentences, and C be the class of all L-structures satisfying A. On C , we consider a binary relation interpreted as accessibility: for M, N ∈ C , we write M N if M accesses N. This gives (C , ) the structure of a Kripke frame whose Kripke models we can study. The natural analysis of such structures is in terms of their modal logic or their structure modal logic (precise definitions of these notions will be given in § 2).
In this paper, we shall consider the class A of abelian groups and the accessibility relation of being (isomorphic to) a subgroup (i.e., A B if and only if A ≤ B). We prove that the modal logic of this structure is the wellknown modal logic S4.2 (Main Theorem 3.1).
Background and related work. Modal logics of classes of structures were originally studied in the context of multiverses of set theory where the modal -operator was interpreted as "in all forcing extensions", "in all ground models", or "in all inner models" to yield the modal logic of forcing, the modal

Definitions and modal logic tools
Modal logic. The language L of modal logic is the closure of a countable set Prop of propositional variables under the binary operation ∧ and the unary operations ¬ and . The symbol is interpreted as "it is necessarily the case that". As usual, we consider ∨, →, ↔, and ♦ to be defined in terms of ∧, ¬, and ; in this case, ♦ is interpreted as "it is possible that". The elements of L are called modal formulas.
A collection Λ of modal formulas closed under modus ponens is called a modal logic; it is called a normal modal logic if it contains the modal formula (p → q) → ( p → q) and is additionally closed under uniform substitution and the necessitation rule (i.e., if ϕ ∈ Λ, then ϕ ∈ Λ; cf. [2, § 1.6]).
Let C be any class and be a definable binary (class) relation on C . We consider (C , ) as a Kripke frame; a valuation is a (class) function v : Prop × C → {0, 1}; the triple (C , , v) is called a Kripke model ; we say that the Kripke model (C , , v) lives on the Kripke frame (C , ).
We can now define Kripke semantics for the language L . If M ∈ C , then In order to avoid metamathematical issues, we shall consider definable valuations, i.e., valuations v such that there is a set parameter π and a set Vol. 84 (2023) The modal logic of abelian groups Page 3 of 12 25 A modal formula ϕ is called definably valid in a Kripke frame (C , ) if it is valid in every definable Kripke model living on (C , ). We write ML(C , ) for the collection of modal formulas definably valid in (C , ), called the modal logic of (C , ). The modal logic relevant for this paper is the modal logic S4.2; it is the smallest normal modal logic containing the formulas Modal logics of classes of structures. Let S be a non-logical vocabulary, L S be the first order language with vocabulary S, and C be a class of L S -structures. Any language L ⊇ L S is called C -adequate if there is a definable model relation |= between elements of C and L-sentences that extends the usual model relation of L S . Examples include infinitary languages or higher-order languages with the vocabulary S. We call a function T : Prop → Sent(L) assigning L-sentences to propositional variables an L-translation. An L-translation gives rise to a valuation for the class C called the L-structure valuation: Since the relation |= is definable, the L-structure valuation is definable. We say that the L-structure modal logic of (C , ) is the set of modal formulas that are valid in each Kripke model (C , , v T ) for an L-translation T . We write ML L (C , ) for the L-structure modal logic of (C , ); by definition and definability, ML(C , ) ⊆ ML L (C , ).
Lower bounds. The validity of modal formulas on frames is closely linked to properties of the relation ; e.g., it is very easy to check that the formula T = p → p is valid on a frame (C , ) if and only if the relation is reflexive. We say that a frame (C , ) is directed if for any M 0 , M 1 ∈ C there is a N ∈ C such that M 0 N and M 1 N. The relevant result for our context is the following theorem. Proof. This is an easy argument along the lines of the standard soundness proofs in modal logic; cf., e.g., [2,Theorems 4.28,and 4.29] and the analogous results for .2 (using directedness).

Upper bounds.
The main tools to determine the structure modal logic of a class C of structures are control statements, as developed in [10,12]. The technique of control statements (based on the technique of Jankov-Fine formulas; cf. [2,  pp. 143sq]) works in a setting where it is suspected that the established lower bound Λ is a normal modal logic with the finite frame property (i.e., there is a class F of finite frames such that ϕ ∈ Λ is ϕ is valid in all frames in F ). Cf. [3, § 6] for an exposition of the general proof strategy for these proofs.
Let S be a vocabulary, C be a class of L S -structures, L be any C -adequate language, and and a binary relation on C . An L-sentence β is called a button Buttons are statements that can be made necessarily true; once they are pushed, they can never become unpushed again; moreover, they are necessarily of that form, i.e., they can necessarily be made necessarily true. In contrast, switches are statements that can always be switched on or off.
We express this formula in words: If exactly the buttons with index in I 0 are pushed and the switches with index in J 0 are switched on, then it is possible to push exactly the buttons with index in I 1 ⊇ I 0 and switch exactly the switches with index in J 1 on.
The existence of large independent sets of buttons and switches is closely linked to getting S4.2 as an upper bound. Remark on the choice of language. Let C be a class of L S -structures for some vocabulary S. If L is a C -adequate language, any L S -translation is a L-translation, and thus we have ML(C , ) ⊆ ML L (C , ) ⊆ ML LS (C , ). As a consequence, if we are in a situation where Theorems 2.1 and 2.2 apply for the first order language L S and we thus obtain then for all C -adequate languages L, the equality S4.2 = ML L (C , ) holds. Therefore, S4.2 = ML LS (C , ) is robust in the sense of [20, p. 1008]. Saveliev and Shapirovsky argue that "intuitively, the robust theory can be considered as a 'true' modal logic of the model-theoretical relation".

The main result
In the following, we shall consider the first-order language of group theory L Gr with a single binary operation symbol + and the usual axioms defining the class A of abelian groups. We use the symbol 0 to denote the neutral element and the symbol − to denote the unary inverse operation; both are definable in L Gr , so we may use them freely in L Gr -formulas. As usual, if n ∈ N and a ∈ A, we define n · a := a + · · · + a n times and −n · a := (−a) + · · · + (−a) n times .
If n ∈ N, we say that an element a ∈ A has order n if n is the least positive number such that n · a = 0; we say that a ∈ A is divisible by n if there is some a * ∈ A such that n · a * = a. For A, B ∈ A , we write A ≤ B if A is a subgroup of B and A × B for the direct product of A and B. As usual, we identify isomorphic groups, so par abus de langage, we use A ≤ B to stand for "A is isomorphic to a subgroup of B".
Clearly, A ≤ A × B and B ≤ A × B. If a has order n or is divisible by n in A and A ≤ B, then a has order n or is divisible by n in B, respectively. Furthermore, if p is a prime number, then A × B has an element of order p if and only if at least one of A and B does.
The relation ≤ is clearly reflexive, transitive, and directed on A . As a consequence of Theorem 2.1, all modal formulas in S4.2 are valid in (A , ≤). The main theorem of this paper is that this lower bound is also an upper bound. Towards a proof of the upper bound, we shall use Theorem 2.2 and produce arbitrarily large independent collections of buttons and switches.

Main
Buttons. If p is a prime number, we define an L Gr -sentence i.e., "there is an element of order p". This is a button: if it is true in any group A, then it will remain true in all B such that A ≤ B. Note that these buttons are so called pure buttons: if one of them is true, then it is pushed (i.e., necessarily true). We can therefore say "β p is pushed in A" if A |= β p . Clearly, the torsion group Z/pZ has the button β p pushed and all others are unpushed. Furthermore, by the above remark about elements of finite order in products, β p is pushed in A × B if and only if it is pushed in A or in B.
These two algebraic facts immediately imply that if P is a finite set of primes, forming the product with the group B P := p∈P Z/pZ will push all buttons with index in P and no additional buttons.
Switches. For our switches, we define for each prime number p the L Grsentence i.e., "every element is divisible by p". It is easy to switch σ p off: if p = q, then for any group A, the group B := A×Z/qZ will have switched σ p off. Switching σ p on requires more work, in particular if we aim to avoid interference with the other switches and buttons. That the sentences σ p are switches will follow from Lemma 3.5.
The following main lemma about these control statements will provide us with all we need to prove the main result. is defined since all values f (a) are integers and only finitely many of them are non-zero. It is easy to see that if f and g are integral, then so are −f and f + g, and (using the fact that A is abelian) we have W (−f ) = −W (f ) and W (f + g) = W (f ) + W (g). For f, g ∈ B A , we say that f and g are equivalent, in symbols, f ∼ g, if f − g is integral and W (f − g) = 0. Note that if f is integral and f ∼ g, then g is integral. Obviously, ∼ is reflexive and since g − f = −(f − g), the above fact about inverses of integral functions shows that it is symmetric. Furthermore,  To see that it is a homomorphism, we need to show for any a, a * ∈ A that [F (a + a * )] = [F (a) + F (a * )]. Since all of the functions F (a) are integral, so are their sums and inverses. We have that In order to show its injectivity, suppose that [F (a)] = [F (a * )], i.e., F (a) − F (a * ) ∼ 0. Thus But this implies that a = a * due to the cancellation laws in groups.

Localisations.
Our second technical tool is that of a localisation. If P is a set of prime numbers, let P be the set of all natural numbers that have only prime factors in P (including 1). We write Q P := { z n ∈ Q ; z ∈ Z, n ∈ P }. By the usual rules of addition of fractions, Q P is an additive subgroup of the rational numbers, i.e., Z ≤ Q P ≤ Q. We shall use the arithmetical fact that for all primes p / ∈ P and a ∈ Q P , we have that a / ∈ Z if and only if p · a / ∈ Z.
Since 0 is integral, p · f must be integral, and hence by (*), f is integral (utilising that p / ∈ P ). Therefore W (f ) is defined and f ∼ 0 implies that W (f ) = 0. But then and hence W (f ) is an element of order p in A.
It is easy to see that the switch σ p is on in Q P if and only if p ∈ P ; this generalises to certain amplifications of groups by Q P . Proof. Firstly, suppose that p ∈ P . Let [f ] ∈ A * [Q P ], i.e., f : A × Z → Q P . This means that for every (a, z) ∈ A × Z, we have that f (a,z) p ∈ Q P . Define Now suppose that p / ∈ P . Consider (0, 1) ∈ A × Z and F (0, 1) ∈ Q P A×Z , i.e., Suppose towards a contradiction that there is some f ∈ Q P A×Z such that p · [f ] = [p · f ] = [F (0, 1)], i.e., p · f ∼ F (0, 1). Since F (0, 1) is integral by definition, this means that p · f must be integral, so ran(p · f ) ⊆ Z and Vol. 84 (2023) The modal logic of abelian groups Page 9 of 12 25 thus, since p / ∈ P , ran(f ) ⊆ Z using (*). By equivalence of p · f and F (0, 1), we obtain (0, 0) = (a,z)∈A×Z (p · f )(a, z) · (a, z) − (0, 1), hence (a,z)∈A×Z (p · f )(a, z) · (a, z) = (0, 1). By the earlier analysis of the integrality of f , we know that the sum on the left-hand side is a finite sum of terms of the form p · ζ · (a, z) for some integer ζ. The group A * = A × Z is a direct product, so we may consider the second coordinate of this sum; we obtain as equation for the second coordinate for some integers ζ i and z i . But then 1 would be divisible by p in Z which is absurd. Contradiction! Note that Lemma 3.5 implies that the sentences σ p are switches: if A is an arbitrary group, then A ≤ A × Z =: A * ≤ A * [Q P ] by Lemma 3.3. By choosing P appropriately, we can create any pattern of switches turned on and off in the group A * [Q P ] by Lemma 3.5.
These technical tools now allow us to finally prove Lemma 3.2.
Proof of Lemma 3.2. Let P and Q be two finite disjoint sets of primes. Let A be an abelian group in which for some P 0 ⊆ P the sentence p∈P0 β p ∧ p∈P \P0 ¬β p holds. Let P 0 ⊆ P 1 ⊆ P and Q 1 ⊆ Q. We need to find a group B such that A ≤ B where the sentence