Starting with classical Galois theory, it has been a common technique in algebra, logic, and other fields of mathematics to investigate structures by means of their automorphisms. This works well for structures with rich automorphism groups; in first-order logic, such models abound: for example, all structures have homogeneous (and highly saturated) elementary extensions, and by the Ehrenfeucht–Mostowski theorem, we can find structures with arbitrarily complex automorphism groups that in a sense control the structures.
On the opposite side of the spectrum, we have structures with only a few automorphisms, and in particular, rigid structures (i.e., having no automorphisms save the identity). Unlike models with many automorphisms, the existence of rigid models is a rather mysterious and whimsical property. On the one hand, many theories have no rigid models at all: this is true of theories as simple as the theory of the infinite set with no further structure, but we can construct arbitrarily complex such theories just by throwing in a new function symbol denoting a nontrivial automorphism. On the other hand, many theories of interest do have rigid models, even arbitrarily large.
For example, there are rigid linear orders (e.g., well orders), and consequently, rigid structures in any class that is “sufficiently universal” so that it can suitably represent all other relational structures (e.g., the theory of graphs). What is more surprising is that there are rigid dense linear orders or Boolean algebras of any uncountable cardinality (cf. ).
Interestingly, some theories have a limited amount of certain “trivial” rigid models, but also many other rigid models for non-trivial reasons. For instance, any completion of Peano arithmetic has a unique prime model where each element is definable; these models are clearly rigid. But in fact, Peano arithmetic has many more rigid models: every model of has a rigid elementary end-extension of the same cardinality (see Kossak and Schmerl [3, Thm. 3.3.14]). For an example involving a more tame theory: archimedean real-closed fields (or any archimedean ordered fields, for that matter) are rigid for trivial reasons, but as proved by Shelah , it also follows from certain combinatorial principles consistent with set theory that there are large, non-archimedean real-closed fields.
In this note, we will have a look at another tame theory: Presburger arithmetic, or equivalently, the theory of -groups. Models of Presburger arithmetic with rich automorphism groups were studied by Llewellyn-Jones . We will instead have a look at rigid models of the theory; in fact, we will present their complete description. We will see that there are, in a sense, both “trivial” and “non-trivial” rigid models, but the amount of non-triviality is limited, which manifests in a cardinality bound: Presburger arithmetic only has rigid models of sizes up to . We also discuss the simpler case of “unordered Presburger arithmetic” , which turns out to have only “trivial” rigid models.
We note that the tamest theory of ordered abelian groups, namely of the divisible ones, has no rigid models (e.g., is always an automorphism). The theory of -groups, which are just extensions of by divisible ordered groups, can thus be thought of as the first nontrivial case in these parts.
In this section, we will review the necessary background information about -groups, and ordered groups in general. For references, basic properties of Presburger arithmetic and -groups are discussed in Prestel and Delzell [6, §4.1], Marker [5, §3.1], and Llewellyn-Jones [4, §2–5]; in a more general set-up, model theory of modules and abelian groups is introduced in Hodges [2, §A.1–2].
In this paper, all groups are commutative, and all ordered structures are linearly ordered.
We assume familiarity with basic properties of ordered groups. We specifically mention that a group can be ordered iff it is torsion-free; more generally, if is a torsion-free group, and an ordered subgroup, the order on extends to an order on . A subgroup of an ordered group is convex if and imply . If is a convex subgroup of , the order on induces an order on . Conversely, if is a subgroup of , is an order on , and an order on , then there exists a unique order on that makes a convex subgroup of , and that induces the order on . Since is explicitly defined by
we will call it the lexicographic order induced by and , even though strictly speaking we should reserve this term for the case where is a direct sum of and .
Recall that a subgroup of is pure if for any integer (w.l.o.g. prime), and any element , if for some , then for some . If is torsion free, is defined uniquely, hence we can restate the condition as follows: whenever for some integer . In particular, if is a subgroup of a torsion-free group , then is the least pure subgroup of that includes .
We will use the fact that divisible groups are injective: in particular, a divisible subgroup is always a direct summand of , and more generally, if is a subgroup with trivial intersection with , we can write for some .
Presburger arithmetic is officially the theory of the monoid , but this structure is bi-interpretable with the more convenient ordered group , and we will work exclusively with the latter. (Bi-interpretation preserves the automorphism group, among many other things, hence this change is transparent for our purposes.)
Models of are called -groups, and they may be characterized as discretely ordered groups such that is divisible. Here, an ordered group is discrete if there exists a least positive element, conventionally denoted ; the subgroup generated by , which is the least nontrivial convex subgroup of , gives a canonical embedding of the ordered group in , and denotes the corresponding quotient group.
The theory of -groups admits partial quantifier elimination: any formula is equivalent to a Boolean combination of linear inequalities , where , and congruences , where .
Let be a -group. For any integer , the residue map is a unique group homomorphism such that . These maps combine to a unique group homomorphism such that , where
is the profinite completion of , and denotes the -adic integers.
We will also consider models of the theory , which we will call unordered -groups. If is a torsion-free group with a distinguished element , let denote the subgroup of generated by ; then is an unordered -group iff is divisible, and is not -divisible for any (equivalently: is a pure subgroup of ; equivalently: is torsion-free).
If is a -group with least positive element , then is an unordered -group, and the convexity of ensures that induces an order on the quotient group . Conversely, if is an unordered -group, the torsion-free group can be ordered, and for any order on , the corresponding lexicographic order on that makes convex gives the structure of a -group. That is, any unordered -group expands to a -group, and these expansions are in 1–1 correspondence with orders on .
Quantifier elimination for unordered -groups takes the form that every formula is equivalent to a Boolean combination of linear equalities , and congruences .
The residue maps and can also be defined for unordered -groups just like before.
Quantifier elimination immediately implies a characterization of elementary substructures: if is an unordered -group, a substructure is elementary iff it is itself an unordered -group iff it is a pure subgroup of . We will call such substructures -subgroups of . Likewise, if is a -group, its elementary substructures are its -subgroups, and these are exactly the substructures containing that are -groups.
3 Leibnizian models
What structures can be said to be rigid for trivial reasons? One case that immediately springs to mind are pointwise definable models: i.e., structures such that every element is definable in without parameters. Clearly, a complete theory may have only one pointwise definable model up to isomorphism (if at all). In our case, the unique pointwise definable -group is the standard model . We are, however, more interested in non-standard (i.e., non-archimedean) examples.
We may, in fact, loosen the condition a bit: in the obvious argument that pointwise definable structures are rigid, we do not really need that each element be isolated by a single formula from the rest of the model—it is enough if we can tell apart any pair of distinct elements. Thus, we are led to the class of pointwise type-definable models; following Enayat , we will call them more concisely Leibnizian models, as they are models that validate one reading of Leibniz’s law of the identity of indiscernibles.
A structure is Leibnizian if for any , there exists a formula (without parameters) such that and . In other words, distinct elements have distinct parameter-free -types.
For example, a real-closed field is Leibnizian if and only if it is archimedean. (This follows easily from quantifier elimination: in a non-archimedean real-closed field, any two infinitely large elements have the same type.) Notice that a Leibnizian structure in a countable language must have cardinality at most . An elementary substructure of a Leibnizian model is Leibnizian.
þ Leibnizian structures are rigid.
Now, how do Leibnizian models of Presburger arithmetic look like? Curiously, the answer does not depend on availability of the order.
þ For any -group , the following are equivalent:
The unordered -group is Leibnizian.
The residue map is an embedding.
This is clear for formulas of the form , as by the choice of . As for linear inequalities, they become somewhat degenerate in one variable: is equivalent to if , and to if . Thus, all nonstandard elements of the same sign (such as and ) satisfy the same inequalities.
Thus, any Leibnizian -group embeds in ; to see that there are many such groups, we only need to observe that itself already works:
þ is an unordered -group.
Proof: is torsion-free as each is a domain of characteristic . Notice that for any prime , an element is -divisible iff . On the one hand, this ensures that is not -divisible; on the other hand, for any , one of is.
Up to isomorphism, Leibnizian unordered -groups are exactly the -subgroups of .
Up to isomorphism, Leibnizian -groups are exactly the -subgroups of , where is a lexicographic order induced by an order on .
In particular, for any , there are non-archimedean Leibnizian -groups and unordered -groups of cardinality .
Notice that distinct -subgroups of are non-isomorphic, as they are the images of their own residue maps.
4 Non-Leibnizian models
As shown by the example of real-closed fields mentioned in Section 1, non-Leibnizian rigid models may be more elusive than Leibnizian models. Before we get to them, we first need to know a little about the general structure of non-Leibnizian models of Presburger arithmetic, or more precisely, of unordered -groups.
þ Let be an unordered -group. We can write , where is a divisible subgroup of , and is a Leibnizian -subgroup of .
Proof: Put as indicated. If and , we have in particular , hence for some . Then , hence as is torsion-free. Thus, is divisible.
Since is divisible, and , we have for some subgroup such that . Since direct summands are necessarily pure, this makes a -subgroup of , and it is Leibnizian as .
Thus, (or its dimension as a -linear space, if we want it numeric) can serve as a measure of non-Leibnizianity of .
In a decomposition with divisible and a Leibnizian -subgroup, is uniquely determined as , and also as the largest divisible subgroup of . On the other hand, is not unique as a subgroup of , though it is of course unique up to isomorphism as .
We stress that the direct sum decomposition in þ4.1 only works at the level of groups; if is ordered, there is no telling how the orders of and interact.
With no order to complicate matters, þ4.1 lends itself to an easy description of automorphisms. But first a bit of notation:
If is an unordered -group, and a subgroup, we will write for . If moreover , , is a homomorphism such that , let be the induced homomorphism .
þ Let be an unordered -group, and and as in þ4.1. Since , the natural quotient map is a group isomorphism of to .
Any automorphism of induces an automorphism of the group such that for all . Such an can be represented in a unique way as
where is an automorphism of the group , and is a group homomorphism with .
Conversely, any as above is induced by a unique automorphism of , namely
We stress that in the statement above, the language of the structure includes the constant , hence it is fixed by all automorphism (whereas and are just group automorphisms).
Proof: An automorphism of has to fix pointwise, hence it indeed induces a group homomorphism , which is in fact an automorphism as induces its inverse. Moreover, must preserve congruence formulas, i.e., the residue map ; thus, for all , and likewise for .
Since , can be uniquely represented as , where , and . The condition further ensures that , and that is a homomorphism . If denotes the natural isomorphism of to , define and by and . Then and . Since is an automorphism, so is .
The converse direction is likewise easy to check.
þ If is a -group, then is an automorphism of if and only if it is an automorphism of the unordered -group , and it is order-preserving. The latter holds if and only if the induced group automorphism is order-preserving, as is a convex subgroup fixed by .
þ The only rigid unordered -groups are the Leibnizian ones.
Proof: If is a non-Leibnizian unordered -group, the divisible torsion-free group in the decomposition is nontrivial. But then has a nontrivial automorphism , for instance . By þ4.4, this lifts to a nontrivial automorphism of , using, e.g., .
This was a rather anticlimactic answer. However, we will see that the situation for ordered -groups is more interesting: some non-Leibnizian -groups are rigid after all. The reason is that some of the automorphisms provided by þ4.4 are not order-preserving. Even so, rigidity severely constraints and , as we will see in a short while.
We will work for a while with divisible subgroups of . Recall that a torsion-free divisible group is nothing else than a -linear space, hence we may treat it with methods from linear algebra. Also notice that a convex subgroup of a divisible ordered group is divisible.
þ Let be a rigid -group, and . Then has no proper convex subgroup such that .
Proof: Assume for contradiction that is such a subgroup. By linear algebra, has a linear subspace such that , which implies . Since is convex, this is not just a direct sum of groups, but also a lexicographic product of the corresponding orders. Thus,
defines an order-preserving automorphism of , which is nontrivial as . Since for all , lifts to a (still order-preserving) automorphism of by þLABEL:lem:aut-unord,rem:unord-ord.
þ Let be a rigid -group, and . If is a convex subgroup of nontrivially intersecting , then .
Proof: If not, then and are nontrivial -linear spaces, hence there exists a nonzero group homomorphism that vanishes on . Define a group homomorphism by . Clearly, ; since , this means has an inverse homomorphism , hence it is an automorphism of . It is also order-preserving: assume that . If , then implies , and a fortiori . If , then . Thus, lifts to a nontrivial automorphism of using þLABEL:lem:aut-unord,rem:unord-ord.
þ Let be a -group, and write with divisible and a Leibnizian -subgroup of as in þ4.1. If is rigid, the following conditions hold:
is cofinal in , or trivial.
is cofinal in .
(i) and (ii): We again identify with . If is positive, let be the least convex subgroup of such that . Then by þ4.8, hence by þ4.7. Thus, integer multiples of are cofinal in (and in after lifting back), including .
Notice that in þ4.9, embeds in , and , being archimedean, embeds in :
þ All rigid -groups have cardinality at most .
þ4.9 is not yet the end of the story, as the given conditions are only necessary, not sufficient. The missing condition is of a rather different (geometric) flavour, putting restrictions on how and sit next to each other inside . On the one hand, it does not seem very illuminating, and on the other hand, it tends to be satisfied for typical examples that one comes up with in practice (as we will see shortly). But anyway, we formulate it here for completeness.
þ Let be a -group, written as as in þ4.1. Assume that is archimedean, and and are both cofinal in .
Let be the largest convex subgroup of such that . By assumption, the ordered group is archimedean, hence there is a homomorphism of ordered groups with , unique up to multiplication by a positive real constant. Put , , and .
The following are equivalent:
The only such that and is .
Proof: Notice that since , restricted to is an isomorphism of to . Let denote its inverse.
Then is a group homomorphism, and for each . We also have and , hence we can apply the same construction to ; a straightforward calculation shows that is the inverse of . Thus, is a group automorphism.
In order to see that it is order-preserving, let . Notice that . Thus, if , then , and a fortiori . If , then , hence as well.
Thus, is an automorphism of , hence it is the identity. But then .
(ii)(i): Let be an automorphism of . By þ4.4, restricts to an automorphism of . Pick any nonzero , and put . Since is order-preserving, and is dense in , we must have for all ; in particular, . More generally, the density of (along with convexity of ) implies that for any , , thus . Since by þ4.4, we obtain . By (ii), . Thus, for all , i.e., is the identity automorphism.
þ Recall that the subgroup in the decomposition is not uniquely determined. In the situation of þ4.11, we may assume without loss of generality that is chosen so that it includes the group defined in the statement of the theorem, as . This makes the choice somewhat more canonical.
(i): Either condition ensures . However, if is such that , then is a -linear embedding of in .
(ii): If is such that , then for some nonzero , thus , and .
þ In order to see that the condition is not automatic, let , and . Clearly, and are -linear subspaces of , and satisfies and . That amounts to for any , which follows from the transcendence of .
While all these classification results are entertaining, we have yet to show the existence of a single non-Leibnizian rigid -group. We are going to remedy this now. First, a simple example.
þ Let be a proper subfield of of cardinality , and for some , so that . By þ4.13, and satisfy condition (ii) of þ4.11. Since , we may fix a group isomorphism , and use it to lift the order from to . This induces a lexicographic order on which makes it a Leibnizian -group. We put as a group, and order it lexicographically so that using . Then is a non-Leibnizian rigid -group by þ4.11.
be a -subgroup of other than ;
be a proper -subgroup of ;
be an order on the group ;
Then there exists a rigid -group with direct decomposition as in þ4.1, and largest convex subgroup trivially intersecting , such that
as a group, is isomorphic to ;
is isomorphic to , equipped with the lexicographic order induced by ;
the ordered groups and are isomorphic to and , respectively.
Proof: We endow with the lexicographic order induced by , making a -group. Since , we may fix a group isomorphism of to , which allows us to transfer the archimedean order from to . We endow with the lexicographic order induced by the already constructed orders on and , making a -group with a convex subgroup .
Put as a group. We equip with the order induced by the fixed isomorphism to , and with the lexicographic order induced by the orders on and , making convex. (This is compatible with the previously constructed order on .)
By construction, is a -group satisfying the listed properties, and by þ4.11, it is rigid.
þ For any infinite cardinals , there are non-Leibnizian rigid -groups such that and , where are as in þ4.1.
þ If our goal were to describe the composition of non-Leibnizian rigid -groups uniquely up to isomorphism, the data used in þ4.16 would not be quite satisfactory: on the one hand, one such -group can be described in several different ways because is not uniquely determined; on the other hand, the given data do not describe a unique -group as we left unspecified the isomorphism of to .
We may improve the description as follows: we replace with (a -subspace of such that and satisfying an appropriate version of þ4.11 (ii)), and a group isomorphism of to . Then any such data describes a unique non-Leibnizian rigid -group up to isomorphism, and conversely, any non-Leibnizian rigid -group is described by almost unique data, the remaining ambiguity being that (along with its subspace ) is only defined up to multiplication by a real constant.
I wish to thank Ali Enayat, whose question prompted the investigation leading to this paper. I would also like to thank Roman Kossak for interesting comments, and the anonymous referee for suggestions that helped to improve the clarity of the paper.
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