Conditions forcing the existence of relative complements in lattices and posets

TL;DR

This paper explores conditions under which relative complements exist in lattices and posets, weakening traditional assumptions and extending the results to more general structures using modular triples.

Contribution

It introduces the concept of modular triples to replace modularity in lattices and extends the theory of relative complements to unbounded posets.

Findings

Modular triples suffice for the existence of relative complements.

Results extend to unbounded posets.

Weaker assumptions replace modularity in lattice theory.

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.