# Distributive lattices in o-minimal structures

**Authors:** Zoltan A. Kocsis  
**arXiv:** [2606.19506](https://arxiv.org/abs/2606.19506)  
**Date:** 2026-06-17  
**Summary page:** https://www.summarizepaper.com/fr/paper/2606.19506v1/

## Abstract

We investigate distributive lattices and Heyting algebras definable in o-minimal structures. We give a complete description of one-dimensional bounded distributive lattices definable over an o-minimal structure expanding a real-closed field, and prove a definable analogue of Birkhoff representation, which we use to classify all one-variable equations in the language of Heyting algebras with respect to whether they can be satisfied in a maximal-dimension subset of a given algebra.