Operators on complemented posets

TL;DR

This paper explores the properties of the complement operator in complemented posets, introduces new binary operators, and characterizes orthogonality relations, including in the Dedekind-MacNeille completion.

Contribution

It studies the operator ^+ in complemented posets, defines new operators, and characterizes orthogonality, including in the Dedekind-MacNeille completion, with an example of a non-Boolean complemented poset.

Findings

^+ can be involutive or antitone in certain posets

Four new binary operators are defined and analyzed

Orthogonality is characterized in the Dedekind-MacNeille completion

Abstract

Given a complemented poset P, we can assign to every element x of P the set x^+ of all its complements. We study properties of the operator ^+ on P, in particular, we are interested in the case when x^+ forms an antichain or when ^+ is involutive or antitone. We apply ^+ to the set Min U(x,y) of all minimal elements of the upper cone U(x,y) of x,y and to the set Max L(x,y) of all maximal elements of the lower cone L(x,y) of x,y. By using ^+ we define four binary operators on P and investigate their properties that are close to adjointness. We present an example of a uniquely complemented poset that is not Boolean. In the last section we study the orthogonality relation induced by complementation. We characterize when two elements of the Dedekind-MacNeille completion of P are orthogonal to each other. Finally, we extend the orthogonality relation from elements to subsets and we prove that two non-empty subsets of P are orthogonal to each other if and only if their convex hulls are orthogonal to each other within the poset of all non-empty convex subsets of P.