Related papers: A nondefinability result for expansions of the ord…

Let $\Omega$ be a complex lattice which does not have complex multiplication and $\wp=\wp_\Omega$ the Weierstrass $\wp$-function associated to it. Let $D\subseteq\mathbb{C}$ be a disc and $I\subseteq\mathbb{R}$ be a bounded closed interval…

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated to sequences such as $(-n^s)_{n>0}$ (for $s>0$) and $(-s^n)_{n>0}$…

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and…

In this article we formally define and investigate the computational complexity of the Definability Problem for open first-order formulas (i.e., quantifier free first-order formulas) with equality. Given a logic $\mathbf{\mathcal{L}}$, the…

We investigate the complexity of the partial order relation of Young's lattice. The definable relations are characterized by establishing the maximal definability property modulo the single automorphism given by conjugation; consequently,…

Consider a linear ordering equipped with a finite sequence of monadic predicates. If the ordering contains an interval of order type \omega or -\omega, and the monadic second-order theory of the combined structure is decidable, there exists…

Suppose that $\Omega$ is a lattice in the complex plane and let $\sigma$ be the corresponding Weierstrass $\sigma$-function. Assume that the point $\tau$ associated to $\Omega$ in the standard fundamental domain has imaginary part at most…

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…

We specify the frontier of decidability for fragments of the first-order theory of ordinal multiplication. We give a NEXPTIME lower bound for the complexity of the existential fragment of $\langle \omega^{\omega^\lambda}; \times, \omega,…

Suppose that $\widetilde{\mathbb R}$ is an o-minimal expansion of the real field in which restricted power functions are definable. We show that if $\widehat{\mathbb R}$ is both a reduct (in the sense of definability) of the expansion…

In development of the started activity on lattice analogues of $W$-algebras, we define the notion of lattice $W_{\infty}$-algebra, accociated with lattice integrable system with infinite set of fields. Various kinds of reduction to lattice…

For a finite real reflection group $W$ with Coxeter element $\gamma$ we give a uniform proof that the closed interval, $[I, \gamma]$ forms a lattice in the partial order on $W$ induced by reflection length. The proof involves the…

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…

An expansion of a definably complete field either defines a discrete subring, or the image of a definable discrete set under a definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation…

We use the Cauchy-Crofton formula to show that every definable cell (bounded by a ball with rational radius) in an O-minimal expansion of a field extension of the real numbers satisfies the Whitney arc property.

We explore "semibounded" expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle R, <, +, \dots\rangle$ is a semibounded o-minimal structure and…

Let $\Lambda\subseteq\mathbb{R}^n$ be a lattice and let $Z\subseteq\mathbb{R}^{m+n}$ be a definable family in an o-minimal expansion of the real field, $\overline{\mathbb{R}}$. A result of Barroero and Widmer gives sharp estimates for the…

We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves structure theorems about their…

Every bounded definable open set is a union of finitely many open strong cells in a weakly o-minimal expansion of a real closed field. We prove this fact and another theorem similar to it.

We prove the following theorem: let $\widetilde{\mathcal R}$ be an expansion of the real field $\overline{\mathbb R}$, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a "semialgebraic…