The operator of relative complementation

TL;DR

This paper investigates the properties and conditions of the operator of relative complementation in lattices, extending known results beyond modular and complemented lattices, and explores its involutive nature and equivalence of different complements.

Contribution

It generalizes the conditions under which a lattice's complement induces a relative complement, and analyzes the operator's properties in various lattice structures.

Findings

Characterizes when a complement induces a relative complement in a lattice.

Identifies conditions for the operator of relative complementation to be involutive.

Provides criteria for different complements to induce the same relative complement.

Abstract

By the operator of relative complementation is meant a mapping assigning to every element x of an interval [a,b] of a lattice L the set x^{ab} of all relative complements of x in [a,b]. Of course, if L is relatively complemented then x^{ab} is non-empty for each interval [a,b] and every element x belonging to it. We study the question under what condition a complement of x in L induces a relative complement of x in [a,b] It is well-known that this is the case provided L is modular and complemented. However, we present a more general result. Further, we investigate properties of the operator of relative complementation, in particular in the case when the interval [a,b] is a modular sublattice of L or if it is finite. Moreover, we characterize when the operator of relative complementation is involutive and we show a class of lattices where this identity holds. Finally, we establish sufficient conditions under which two different complements of a given element x of [a,b] induce the same relative complement of x in this interval.