# Residuation in lattice effect algebras

## TL;DR

This paper establishes a correspondence between lattice effect algebras and quasiresiduated lattices, providing a new structural perspective and reconstruction method for these algebraic systems, including pseudoeffect algebras.

## Contribution

It introduces quasiresiduated lattices and demonstrates their equivalence and reconstructive relationship with lattice effect algebras and pseudoeffect algebras.

## Key findings

- Every lattice effect algebra can be organized into a commutative quasiresiduated lattice.
- Every such lattice can be converted back into a lattice effect algebra.
- Good lattice pseudoeffect algebras can be organized into quasiresiduated lattices with divisibility.

## Abstract

We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such a lattice can be converted into a lattice effect algebra and every lattice effct algebra can be reconstructed form its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced recently by Dvurecenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such a lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.