The operator of relative complementation

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 xab of all relative complements of x in [a, b]. Of course, if L is relatively complemented then xab is nonempty 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 ab of relative complementation satisfies the identity (xab)ab ≈ x provided [a, b] is complemented 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.