Let $L$ is a lattice.
At PlanetMath 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?
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 $\le$.