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On Lattice Complements

Published online by Cambridge University Press:  18 May 2009

Robert Bumcrot
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 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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 7 , Issue 1 , January 1965 , pp. 22 - 23
Copyright
Copyright © Glasgow Mathematical Journal Trust 1965

References

REFERENCES

1.Birkhoff, G., Lattice theory (American Mathematical Society Colloquium Publications, Vol. 25; 2nd edition, 1948).Google Scholar
2.Dilworth, R. P., Lattices with unique complements, Trans. Amer. Math. Soc. 57 (1945), 123–154.CrossRefGoogle Scholar