Complemented Lattices
William Elliot
Let L be a complemented lattice. That is a lattice with
a top or max element 1 and a bottom or min element 0 and
for all x in L, some y in L with
xy = inf x,y = 0, x + y = sup x,y = 1
x and y with that property are call complements.
If L is a distributive lattice, then each element has
exactly one complement. Is the converse correct,
that if L is complemented lattice for which every element
has a unique complement, then L is distributive? How does
one go about proving that?
----
tc...@lsa.umich.edu
In article <garnlm$5gc$1...@dizzy.math.ohio-state.edu>,
William Elliot <ma...@hevanet.remove.com> wrote:
>If L is a distributive lattice, then each element has
>exactly one complement. Is the converse correct,
>that if L is complemented lattice for which every element
>has a unique complement, then L is distributive? How does
>one go about proving that?
Yes; for example Corollary 1 in Chapter IX of Birkhoff's book on lattice
theory says that a lattice is distributive if and only if relative
complements in it are uniquely determined. He proves it via a structure
theorem: If a lattice fails to be distributive then it must contain one of
three specific sublattices, and you can construct non-unique complements
in each of these three cases.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
Zundark
> Is the converse correct, that if L is complemented lattice for which
> every element has a unique complement, then L is distributive?
No. (I see that Tim Chow has given the opposite answer. However, the
theorem he quotes uses a stronger condition - the existence of unique
relative complements.)
R. P. Dilworth proved in 1945 that every lattice can be embedded in a
uniquely complemented lattice. So a uniquely complemented lattice need
not be distributive, or even modular. This result and much more can be
found in a book by V. N. Salii:
V. N. Salii, Lattices with Unique Complements,
Translations of Mathematical Monographs, Volume 69,
American Mathematical Society, Providence, 1988.
The following review of Salii's book gives some idea of the complexity of
uniquely complemented lattices:
http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0605.06009&format=complete
tc...@lsa.umich.edu
Zundark <zun...@zundark.invalid> wrote:
>William Elliot wrote:
>
>> Is the converse correct, that if L is complemented lattice for which
>> every element has a unique complement, then L is distributive?
>
>No. (I see that Tim Chow has given the opposite answer. However, the
>theorem he quotes uses a stronger condition - the existence of unique
>relative complements.)
You are correct...I misunderstood the original question.