Galois connections and isomorphism of simultaneous ordered relations

TL;DR

This paper introduces the concept of binary posets, studying partially ordered sets under two relations simultaneously, and explores their duality, maximal/minimal elements, and Galois connections.

Contribution

It extends order theory by defining binary posets and establishing their duality, along with new theorems on order isomorphism and Galois connections.

Findings

Binary posets follow the duality principle like posets.

New definitions for maximal and minimal elements in binary posets.

Theorems on order isomorphism and Galois connections for binary posets.

Abstract

In order theory, partially ordered sets are only equipped with one relation which decides the entire structure/Hasse diagram of the set. In this paper, we have presented how partially ordered sets can be studied under simultaneous partially ordered relations which we have called binary posets. The paper is motivated by the problem of operating a set simultaneously under two distinct partially ordered relations. It has been shown that binary posets follow the duality principle just like posets do. Within this framework, some new definitions concerning maximal and minimal elements are also presented. Furthermore, some theorems on order isomorphism and Galois connections are derived.