You are here
Homeuniquely complemented lattice
Primary tabs
uniquely complemented lattice
Recall that in a bounded distributive lattice, complements, relative complements, and differences of lattice elements, if exist, must be unique. This leads to the general consideration of general bounded lattices in which complements are unique.
Definition. A complemented lattice such that every element has a unique complement is said to be uniquely complemented. If $a$ is an element of a uniquely complemented lattice, $a^{{\prime}}$ denotes its (unique) complement. One can think of ${}^{{\prime}}$ as a unary operator on the lattice.
One of the first consequences is
$a^{{\prime\prime}}=a.$ 
To see this, we have that $a\vee a^{{\prime}}=1$, $a\wedge a^{{\prime}}=0$, as well as $a^{{\prime\prime}}\vee a^{{\prime}}=1$, $a^{{\prime\prime}}\wedge a^{{\prime}}=0$. So $a=a^{{\prime\prime}}$, since they are both complements of $a^{{\prime}}$.
Below are some additional (and nontrivial) properties of a uniquely complemented lattice:

there exists a uniquely complemented lattice that is not distributive

a uniquely complemented lattice $L$ is distributive if at least one of the following is satisfied:
(a) ${}^{{\prime}}$, as an operator on $L$, is order reversing;
(b) $(a\vee b)^{{\prime}}=a^{{\prime}}\wedge b^{{\prime}}$;
(c) $(a\wedge b)^{{\prime}}=a^{{\prime}}\vee b^{{\prime}}$;
(d) (von Neumann) $L$ is a modular lattice;
(e) (BirkhoffWard) $L$ is an atomic lattice.
In fact, the first three conditions are equivalent, so that $L$ is distributive if it satisfies the de Morgan’s laws.

(Dilworth) every lattice can be embedded in a uniquely complemented lattice.
References
 1 T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005).
 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
Mathematics Subject Classification
06B05 no label found06C15 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Prime numbers out of sequence by Rubens373
Oct 7
new question: Lorenz system by David Bankom
Oct 19
new correction: examples and OEIS sequences by fizzie
Oct 13
new correction: Define Galois correspondence by porton
Oct 7
new correction: Closure properties on languages: DCFL not closed under reversal by babou
new correction: DCFLs are not closed under reversal by petey
Oct 2
new correction: Many corrections by Smarandache
Sep 28
new question: how to contest an entry? by zorba
new question: simple question by parag