Residuation in finite posets

TL;DR

This paper explores residuation in finite posets, generalizing implications and conjunctions beyond lattice structures, with simplified formulas and illustrative examples for better understanding.

Contribution

It introduces a generalized operator residuation framework for finite posets, extending previous work on quantum logic structures and simplifying related formulas.

Findings

Operators can be defined using maximal or minimal elements in finite posets.

Formulas for implication and conjunction are simplified in finite posets.

Theoretical results are supported by illustrative examples.

Abstract

When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce the connective implication to be everywhere defined and satisfying (left) adjointness with the connective conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.