# Error at PlanetMath? (lattice theory: pseudocomplement)

Let $L$ is a lattice.

At http://planetmath.org/encyclopedia/Benzene.html it's written:

It is easy to see that given an element $a\in L$, the pseudocomplement of $a$, if it exists, is unique.

Is it true in general? I see a proof only for special classes of lattices, such as distributive lattice. Is it an error in PlanetMath or I just miss a proof for the general case?

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This follows directly from the properties of the pseudocomplement given above the quoted sentence. If $b$ and $b'$ are pseudocomplements of an element $a$, then property 1 says that $b\land a=0$ and $b'\land a=0$. Then property 2 of $b$ implies $b'\le b$, and property 2 of $b'$ implies $b\le b'$. This implies $b=b'$ by the antisymmetry of the partial order $\le$.