The theory of $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ is decidable if $a$ is quadratic. If $a$ is the golden ratio, $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ defines multiplication by $a$. The results are established by using the Ostrowski numeration system based on the continued fraction expansion of $a$ to define the above structures in monadic second order logic of one successor. The converse that $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ defines monadic second order logic of one successor, will also be established.
Given a dense additive subgroup $G$ of $\mathbb R$ containing $\mathbb Z$, we consider its intersection $\mathbb G$ with the interval $[0,1[$ with the induced order and the group structure given by addition modulo $1$. We axiomatize the theory of $\mathbb G$ and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a "standard" part and two ordered semigroups of infinitely small and infinitely large elements.
Let $\mathcal{R}$ be an expansion of the ordered real additive group. When $\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval $I\subseteq \mathbb{R}$, or $\mathcal{R}$ is a reduct of an ordered vector space. We say $\mathcal{R}$ is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of $(\mathbb{R},<,+)$. In particular, we show that for expansions that do not define dense $ω$-orders (we call these type A expansions), an appr…
We prove that for an o-minimal expansion of the real additive group $\cal R$ and a set $P\subseteq \mathbb{R}$ of dimension $0$ such that $\langle\mathcal{R},P\rangle$ is sparse, has definable choice and every definable set has interior or is nowhere dense then, for every definable set $X$, there is a family $\{X_t:\; t\in A\}$ definable in \Cal R and a set $S\subseteq A$ of dimension $0$ such that $X=\bigcup_{t\in S}X_t$. Moreover, in the d-minimal setting, there is a finite decomposition of $X$ into sets of the previous form such that for every $t\in S$ $X_t$ is relatively open in $\bigcup_{…
We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For $\mathcal{R}=( \mathbb{R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq \mathbb{R}$ a set satisfying certain tameness conditions, we discuss under which conditions $(\mathcal R,P)$ defines total linear functions that are not definable in \mathcal{R}. Examples of such structures that does def…
Let $\mathcal{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathcal{R}$ is generically locally o-minimal. It follows that if $X \subseteq \mathbb{R}^n$ is definable in $\mathcal{R}$ then the $C^k$-points of $X$ are dense in $X$ for any $k \geq 0$. This follows from a more general theorem on $\mathrm{NIP}$ expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable and $\bigwedge$-d…
We study an extension of \g propositional logic whose corresponding algebra is an ordered Abelian group. Then we expand the ideas to first-order case of this logic.
In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $Γ/ Δ$ for each definable convex subgroup $Δ$, and sorts for the quotient groups $Γ/ Δ+ lΓ$ where $Δ$ is a definable convex subgroup and $l \in \mathbb{N}_{\geq 2}$. We refer to these sorts as the \emph{quotient sorts}. For the dp-minimal case we obtain a complete elimination of imaginaries, if we also add constants …
If $\mathcal{Z}$ is a dp-minimal expansion of a discrete ordered abelian group $(Z,<,+)$ and $\mathcal{Z}$ does not admit a nontrivial definable convex subgroup then $\mathcal{Z}$ is interdefinable with $(Z,<,+)$ and $(Z,<,+)$ is elementarily equivalent to $(\mathbb{Z},<,+)$.
By a result known as Rieger's theorem (1956), there is a one-to-one correspondence, assigning to each cyclically ordered group $H$ a pair $(G,z)$ where $G$ is a totally ordered group and $z$ is an element in the center of $G$, generating a cofinal subgroup $\langle z\rangle$ of $G$, and such that the quotient group $G/\langle z\rangle$ is isomorphic to $H$. We first establish that, in this correspondence, the first order theory of the cyclically ordered group $H$ is uniquely determined by the first order theory of the pair $(G,z)$. Then we prove that the class of cyclically ordered groups is a…
The Baer-Specker group is the product of countably many copies of the additive group Z of integers. Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to Z.
In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field No of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of No, i.e. a subfield of No that is an initial subtree of No. In this sequel to [15], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of No that are themselves initial. It is further show…
We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly ordered set $Ω$, there are uncountably many maximal subgroups of $\operatorname{End}(\mathbb{Q},{\leq})$ isomorphic to the automorphism group of $Ω$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as a retract in $(\mathbb{Q},{\leq})$ in terms of topological information concerning $X$. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal…
We determine the sets definable in expansions of the ordered real additive group by generalized Cantor sets. Given a natural number $r\geq 3$, we say a set $C$ is a generalized Cantor set in base $r$ if there is a non-empty $K\subseteq\{1,\ldots,r-2\}$ such that $C$ is the set of those numbers in $[0,1]$ that admit a base $r$ expansion omitting the digits in $K$. While it is known that the theory of an expansion of the ordered real additive group by a single generalized Cantor set is decidable, we establish that the theory of an expansion by two generalized Cantor sets in multiplicatively inde…
We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion.
This paper provides a full characterization for when the expansion of a complete o-minimal theory by a unary predicate that picks out a divisible dense and codense subgroup has a model companion. This result is motivated by criteria and questions introduced in the recent works concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. The focus of this paper is establishing the companionability dividing line in the o-minimal setting because this allows us to provide a full and geometric characterization. E…
We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of an algebraic group by a Cartesian power of the group of green elements. A definable subgroup of an algebraic group in the green or red field is an extension of the coloured points of a multiplicative or additive algebraic group by an algebraic group. In particular, a simple group in a coloured field is algebraic.
Given a class C of subgroups of a topological group G, we say that a subgroup H in C is a universal C subgroup of G if every subgroup K in C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respect to a natural pre-order on the set of subgroups of G. We show that for any locally compact Polish group G, the countable power of G has a universal K-sigma subgroup and a universal compactly generated subgroup. We prove a weaker version of this in the non-locally compact case and provide an example showing that this result cannot readily be impro…
We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in CODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorp…
We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.
Universität Zürich（苏黎世大学）；Fachbereich Mathematik und Statistik, Universität Konstanz（康斯坦茨大学数学与统计系）；Institut für Interdisziplinäre Sprachevolutionswissenschaft, Universität Zürich（苏黎世大学跨学科语言进化科学研究所）
We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition. In this note we attempt an introductory investigation of totally (also partially) ordered monoids, not assumed commutative (still writing them additively), and taking them as positive, i.e.\ every element is greater than the unit element. That, in the usual commutative cases, allows the ordering to be defined via the algebraic structure, namely, as divisibil…
In answer to a question of A. Blass, J. Irwin and G. Schlitt, a subgroup G of the additive group Z^ω is constructed whose dual, Hom(G,Z), is free abelian of rank 2^{\aleph_0}. The question of whether Z^ω has subgroups whose duals are free of still larger rank is discussed, and some further classes of subgroups of Z^ω are noted.
We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $ε_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $α< ε_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $α+ 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr \mathbf{Z}$, a…
Probability maps are additive and normalised maps taking values in the unit interval of a lattice ordered Abelian group. They appear in theory of affine representations and they are also a semantic counterpart of Hajek's probability logic. In this paper we obtain a correspondence between probability maps and positive operators between certain Riesz spaces, which extends the well-known representation theorem of real-valued MV-algebraic states by positive linear functionals. When the codomain algebra contains all continuous functions, the set of all probability maps is convex, and we prove that …
By recent work on some conjectures of Pillay, each definably compact group $G$ in a saturated o-minimal expansion of an ordered field has a normal ``infinitesimal subgroup'' $G^{00}$ such that the quotient $G/G^{00}$, equipped with the ``logic topology'', is a compact (real) Lie group. Our first result is that the functor $G\mapsto G/G^{00}$ sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group $G/G^{00}$ and the o-minimal spectrum $\widetilde G$ of $G$. We prove that $G/G^{00}$ is a topological quotient of $\…
