Geodesic
On Lattice Complements
Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 22-23

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Let (L, ≦) be a distributive lattice with first element 0 and last element 1. If a, b in L have complements, then these must be unique, and the De Morgan laws provide complements for a ∧ b and a ∨ b. We show that the converse statement holds under weaker conditions.Theorem 1. If(L, ≦) is a modular lattice with 0 and 1 and if a, b in L are such that a ≦b and a ≨ b have (not necessarily unique) complements, then a andb have complements. 
Bumcrot, Robert. On Lattice Complements. Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 22-23. doi: 10.1017/S2040618500035103
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[1] 1.Birkhoff, G., Lattice theory (American Mathematical Society Colloquium Publications, Vol. 25; 2nd edition, 1948). Google Scholar

[2] 2.Dilworth, R. P., Lattices with unique complements, Trans. Amer. Math. Soc. 57 (1945), 123–154. Google Scholar | DOI

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