Residuated operators in complemented posets

TL;DR

This paper introduces a generalized residuation framework in complemented posets using operators based on upper and lower cones, extending residuation concepts to various poset structures.

Contribution

It defines new operators M and R in posets with unary operations and demonstrates their application to Boolean and pseudo-orthomodular posets, broadening residuation theory.

Findings

Boolean posets can be structured as operator residuated posets.

Pseudo-orthomodular posets can also be organized into operator residuated structures.

Results highlight the analogy between pseudo-orthomodular posets and orthomodular lattices.

Abstract

Using the operators of taking upper and lower cones in a poset with a unary operation, we define operators M(x,y) and R(x,y) in the sense of multiplication and residuation, respectively, and we show that by using these operators, a general modification of residuation can be introduced. A relatively pseudocomplemented poset can be considered as a prototype of such an operator residuated poset. As main results we prove that every Boolean poset as well as every pseudo-orthomodular poset can be organized into a (left) operator residuated structure. Some results on pseudo-orthomodular posets are presented which show the analogy to orthomodular lattices and orthomodular posets.