# Residuated operators and Dedekind-MacNeille completion

## TL;DR

This paper investigates how residuated operators on certain posets can be extended to Dedekind-MacNeille completions, establishing conditions under which these completions form orthomodular lattices, especially in the context of pseudo-orthomodular posets.

## Contribution

It characterizes when Dedekind-MacNeille completions of pseudo-orthomodular posets are orthomodular lattices and introduces the concept of strongly D-continuous pseudo-orthomodular posets.

## Key findings

- Dedekind-MacNeille completion can form a residuated lattice for Boolean and pseudocomplemented posets.
- Conditions are identified for operators to yield residuation in pseudo-orthomodular posets.
- Strongly D-continuous pseudo-orthomodular posets have their completions as orthomodular lattices.

## Abstract

The concept of operator residuation for bounded posets with unary operation was introduced by the first two authors. It turns out that in some cases when these operators are transformed into lattice terms and the poset ${\mathbf P}$ is completed into a Dedekind-MacNeille completion $\BDM(\mathbf P)$ then the complete lattice $\BDM(\mathbf P)$ becomes a residuated lattice with respect to these transformed terms. It is shown that this holds in particular for Boolean posets and for relatively pseudocomplemented posets. More complicated situation is with orthomodular and pseudo-orthomodular posets. We show which operators $M$ (multiplication) and $R$ (residuation) yield operator left-residuation in a pseudo-orthomodular poset ${\mathbf P}$ and if $\BDM(\mathbf P)$ is an orthomodular lattice then the transformed lattice terms $\odot$ and $\to$ form a left residuation in $\BDM(\mathbf P)$. However, it is a problem to determine when $\BDM(\mathbf P)$ is an orthomodular lattice. We get some classes of pseudo-orthomodular posets for which their Dedekind-MacNeille completion is an orthomodular lattice and we introduce the so called strongly $D$-continuous pseudo-orthomodular posets. Finally we prove that,for a pseudo-orthomodular poset ${\mathbf P}$, the Dedekind-MacNeille completion $\BDM(\mathbf P)$ is an orthomodular lattice if and only if ${\mathbf P}$ is strongly $D$-continuous.