# A categorical equivalence for Stonean residuated lattices

## TL;DR

This paper establishes a categorical equivalence between Stonean residuated lattices and triples consisting of their Boolean skeleton, dense elements, and a connecting map, extending previous ideas to this specific algebraic structure.

## Contribution

It introduces a new categorical framework for Stonean residuated lattices using triples, generalizing earlier approaches for related algebraic structures.

## Key findings

- Proves a categorical equivalence between Stonean residuated lattices and triples.
- Defines a category of triples with objects and morphisms.
- Demonstrates applications of the equivalence in algebraic contexts.

## Abstract

Distributive Stonean residuated lattices are closely related to Stone algebras since their bounded lattice reduct is a Stone algebra. In the present work we follow the ideas presented by Chen and Gr\"{a}tzer and try to apply them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. We compare our results with other works and show some applications of the equivalence.