# How to introduce the connective implication in orthomodular posets

## TL;DR

This paper introduces a new form of implication in orthomodular posets, linking it with conjunction through unsharp adjointness, and explores the conditions under which these structures can be converted into each other.

## Contribution

It defines an implication for orthomodular posets that allows constructing unsharp residuated posets and establishes a near equivalence between these structures.

## Key findings

- Defined an implication compatible with orthomodular posets
- Constructed unsharp residuated posets from the implication
- Established conditions for converting between structures

## Abstract

Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be introduced in such a way that it is related with conjunction, i.e. with the partial operation meet, by means of some kind of adjointness. We present here such an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptions, such an unsharp residuated poset can be converted into an orthomodular poset and that this assignment is nearly one-to-one.