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On Lattice Complements

Published online by Cambridge University Press:  18 May 2009

Robert Bumcrot
Affiliation:
Ohio State University Columbus, Ohio
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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 ab and ab. 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.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1965

References

REFERENCES

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2.Dilworth, R. P., Lattices with unique complements, Trans. Amer. Math. Soc. 57 (1945), 123154.CrossRefGoogle Scholar