Conditions forcing the existence of relative complements in lattices and posets

Abstract

It is elementary and well known that if an element x of a bounded modular lattice L has a complement in L then x has a relative complement in every interval [a, b] containing x. We show that the relatively strong assumption of modularity of L can be replaced by a weaker one formulated in the language of so-called modular triples. We further show that, in general, we need not suppose that x has a complement in L . By introducing the concept of modular triples in posets, we extend our results obtained for lattices to posets. It should be remarked that the notion of a complement can be introduced also in posets that are not bounded.