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?


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 $\le$.

  • $\begingroup$ 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 $\endgroup$ – porton Nov 1 '12 at 21:43
  • 1
    $\begingroup$ @porton: Strange, I can't find the word "maximum" on either page -- where did you find it? $\endgroup$ – joriki Nov 1 '12 at 21:49
  • $\begingroup$ Search "maximal element" at planetmath.org/encyclopedia/Benzene.html $\endgroup$ – porton Nov 1 '12 at 21:51
  • 1
    $\begingroup$ @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"? $\endgroup$ – joriki Nov 1 '12 at 21:52
  • 2
    $\begingroup$ @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? $\endgroup$ – joriki Nov 1 '12 at 22:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.