Related papers: Ordered abelian groups that do not have eliminatio…
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…
In this paper we study elimination of imaginaries in some classes of henselian valued fields of equicharacteristic zero and residue field algebraically closed. The results are sensitive to the complexity of the value group. We focus first…
We introduce a formalism of infinite, linearly ordered products in general groups. Using this, we define infinite compositions in certain groups of formal power series such as transseries. We show that such groups can sometimes be…
We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in…
We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is…
We investigate when an ordered abelian group $G$ is stably embedded in a given elementary extension $H$. We focus on a large class of ordered groups which includes maximal ordered groups with interpretable archimedean valuation. We give a…
In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more…
We consider strong expansions of the theory of ordered abelian groups. We show that the assumption of strength has a multitude of desirable consequences for the structure of definable sets in such theories, in particular as relates to…
As mathematical induction is applied to prove statements on natural numbers, {\it continuous induction} (or, {\it real induction}) is a tool to prove some statements in real analysis.(Although, this comparison is somehow an overstatement.)…
Given a finite group $G$, we denote by $\psi\,'(G)$ the product of element orders of $G$. Our main result proves that the restriction of $\psi\,'$ to abelian $p$-groups of order $p^n$ is strictly increasing with respect to a natural order…
We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$…
Let $V$ be a finite abelian group of odd order, equipped with a non-degenerate, alternating form $\omega\colon V\times V \to \mathbb{Z}/m\mathbb{Z}$. We give closed formulas for the character values of the Weil representation associated…
We classify the imaginaries in a large class of equicharacteristic zero henselian valued fields that contain all those with bounded inertia group, and more. To do so, we consider a mix of sorts introduced in earlier works of the two authors…
We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product…
We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the ideal of…
We show that types over real algebraically closed sets are stationary, both for the theory of separably closed fields of infinite degree of imperfection and for the theory of beautiful pairs of algebraically closed field. The proof is given…
In this paper we define Ordered Generating System for finite non-abelian groups, which is a generalization of the basis theorem for finite abelian groups. We prove the following: If each composition factor of a group G has Ordered…
In this paper we study the satisfiability and solutions of group equations when combinatorial, algebraic and language-theoretic constraints are imposed on the solutions. We show that the solutions to equations with length, lexicographic…
We characterize and construct linearly ordered sets, abelian groups and fields that are {\emph symmetrically complete}, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and…
We prove that the category of ordered abelian groups equipped with an automorphism has the Amalgamation Property, deduce that their inductive theory is NIP in the sense of positive logic, and initiate a development of the latter framework.…