Operators on complemented posets
Michal Botur, Ivan Chajda, Helmut Länger
TL;DR
This paper studies the operator $^+$ on bounded complemented posets, where $a^+=\{x\in P\mid x\perp a\}$ collects all complements of $a$, and investigates its interaction with upper and lower cones via $\mathrm{Min}\,U$ and $\mathrm{Max}\,L$ to define four derived binary operators that exhibit adjoint-like properties under various structural assumptions. It establishes that the closed sets under $^+$ form a complete ortholattice and characterizes when $x^+$ forms an antichain or when $(x^+)^+$ is injective, including counterexamples to generalized De Morgan laws in non-uniquely complemented posets. The work further analyzes adjointness of the derived operators under finite-length and modularity conditions, with special attention to Boolean posets where adjointness and De Morgan identities hold. Finally, it connects orthogonality through the Dedekind–MacNeille completion and the poset of non-empty convex subsets, showing that orthogonality in the original poset corresponds to orthogonality of convex hulls in the convex-subset lattice, thereby linking complementation, completion, and convexity in a unified framework.
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.