# 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/en/paper/2606.19506v1/

## Key points

- Study focuses on distributive lattices and Heyting algebras within o-minimal structures
- Research explores one-dimensional bounded distributive lattices over an o-minimal structure expanding a real-closed field
- Establishment of a definable analogue of Birkhoff representation is a key highlight
- Classification of one-variable equations in the language of Heyting algebras is significant for determining satisfaction within a maximal-dimension subset
- Research contributes to deeper understanding of distributive lattices and Heyting algebras within o-minimal structures, highlighting their properties and relationships

## Summary

This study by Zoltan A. Kocsis focuses on distributive lattices and Heyting algebras that can be defined within o-minimal structures. The research delves into one-dimensional bounded distributive lattices that are definable over an o-minimal structure expanding a real-closed field. The key highlight of the investigation is the establishment of a definable analogue of Birkhoff representation, which plays a pivotal role in classifying all one-variable equations in the language of Heyting algebras. This classification is significant as it determines whether these equations can be satisfied within a maximal-dimension subset of a given algebra. Through comprehensive description and analysis, this research contributes to a deeper understanding of distributive lattices and Heyting algebras within the framework of o-minimal structures, shedding light on their intricate properties and relationships.