# 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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## 1 Answer

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 $\le$.

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It seems that in PlanetMath "maximum of a set" and "greatest element of a set" are confused. See planetmath.org/?op=getobj;from=objects;id=2749 –  porton Nov 1 '12 at 21:43
@porton: Strange, I can't find the word "maximum" on either page -- where did you find it? –  joriki Nov 1 '12 at 21:49
Search "maximal element" at planetmath.org/encyclopedia/Benzene.html –  porton Nov 1 '12 at 21:51
@porton: How am I going to find "maximum of a set" by searching for "maximal element"? Or are you saying that your first comment was in error and you meant "maximal element"? –  joriki Nov 1 '12 at 21:52
@porton: I'm getting more and more confused. Why are you talking about "maximum of a set"? I asked where you found this expression on these pages, but you pointed me to "maximal element" instead. What's the relevance of the definition of a term that doesn't appear anywhere on the pages in question? –  joriki Nov 1 '12 at 22:01