Operator residuation in orthomodular posets of finite height

TL;DR

This paper introduces a novel operator residuation framework for finite-height orthomodular posets, enabling their reconstruction and extending to related poset classes, thus advancing the algebraic understanding of quantum logic structures.

Contribution

It defines a new operator residuated structure for orthomodular posets, establishing a near one-to-one correspondence and extending the approach to weakly orthomodular posets.

Findings

Operators form an adjoint pair on orthomodular posets.

Original poset can be recovered from the operator residuated structure.

Applicable to weakly orthomodular and dually weakly orthomodular posets.

Abstract

We show that for every orthomodular poset P of finite height there can be defined two operators forming an adjoint pair with respect to an order-like relation defined on the power set of P. This enables us to introduce the so-called operator residuated poset corresponding to P from which the original orthomodular poset P can be recovered. Moreover, this correspondence is almost one-to-one. We show that this construction of operators can be applied also to so-called weakly orthomodular and dually weakly orthomodular posets. Examples of such posets are included.