2024/09/18 by Ivan Chajda, Chajda, Ivan, Helmut Länger +1
Computer Science · Mathematics · #06A15 #06C05 #06C15 #06C20 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2409.12291
openalex publication_date 2024/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.