Conditions forcing the existence of relative complements in lattices and posets
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Publication:6372577
DOI10.1515/MS-2023-0003arXiv2107.05367MaRDI QIDQ6372577
Publication date: 5 July 2021
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.
Partial orders, general (06A06) Complemented lattices, orthocomplemented lattices and posets (06C15) Algebraic aspects of posets (06A11) Complemented modular lattices, continuous geometries (06C20)
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