Uniquely Complemented Lattice (original) (raw)

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A uniquely complemented lattice is a complemented lattice (L, ^ , v ,0,1,^') that satisfies

( forall x in L)( forall y in L)[(x ^ y=0) ^ (x v y=1)]=>y=x^'.

The class of uniquely complemented lattices is not a subvariety of the class of complemented lattices. On the other hand, there is a well-known class of uniquely complemented lattices that is a subvariety of the variety of complemented lattices, namely the class of Boolean algebras. They form a variety because they are the distributive complemented lattices, and one can prove that any distributive complemented lattice is uniquely complemented.

See also

Complemented Lattice

This entry contributed by Matt Insall (author's link)

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Insall, Matt. "Uniquely Complemented Lattice." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/UniquelyComplementedLattice.html

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