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

Conditions forcing the existence of relative complements in lattices and posets