PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
orthocomplemented lattice (Definition)

An orthocomplemented lattice is a complemented lattice in which every element has a distinguished complement, called an orthocomplement, that behaves like the complementary subspace of a subspace in a vector space.

Formally, let $ L$ be a complemented lattice and denote $ M$ the set of complements of elements of $ L$. $ M$ is clearly a subposet of $ L$, with $ \le$ inherited from $ L$. For each $ a\in L$, let $ M_a\subseteq M$ be the set of complements of $ a$. $ L$ is said to be orthocomplemented if there is a function $ ^{\perp}:L \to M$, called an orthocomplementation, whose image is written $ a^{\perp}$ for any $ a\in L$, such that

  1. $ a^{\perp}\in M_a$,
  2. $ (a^{\perp})^{\perp}=a$, and
  3. $ \perp$ is order-reversing; that is, for any $ a,b\in L$, $ a\le b$ implies $ b^{\perp}\le a^{\perp}$.

The element $ a^{\perp}$ is called an orthocomplement of $ a$ (via $ ^{\perp}$).

Examples. In addition to the example of the lattice of vector subspaces of a vector space cited above, let's look at the Hasse diagrams of the two finite complemented lattices below,

$\displaystyle \xymatrix{ & 1 \ar@{-}[ld] \ar@{-}[d] \ar@{-}[rd] & \\ a \ar@{-}[... ... \ar@{-}[rrd] & b \ar@{-}[rd] & & c \ar@{-}[ld] & d \ar@{-}[lld] \\ & & 0 & & }$    

the one on the right is orthocomplemented, while the one on the left is not. From this one deduces that orthcomplementation is not unique, and that the cardinality of any finite orthocomplemented lattice is even.

Remarks.

  • From the first condition above, we see that an orthocomplementation $ \perp$ is a bijection. It is one-to-one: if $ a^{\perp}=b^{\perp}$, then $ a=(a^{\perp})^{\perp}=(b^{\perp})^{\perp}=b$. And it is onto: if we pick $ a\in M\subseteq L$, then $ (a^{\perp})^{\perp}=a$. As a result, $ M=L$, every element of $ L$ is an orthocomplement. Furthermore, we have $ 0^{\perp}=1$ and $ 1^{\perp}=0$.
  • Let $ L^{\prime}$ be the dual lattice of $ L$ (a lattice having the same underlying set, but with meet and join operations switched). Then any orthocomplementation $ \perp$ can be viewed as a lattice isomorphism between $ L$ and $ L^{\prime}$.
  • From the above conditions, it follows that elements of $ L$ satisfy the de Morgan's laws: for $ a,b\in L$, we have
      $\displaystyle a^{\perp}\wedge b^{\perp}=(a\vee b)^{\perp},$ (1)
      $\displaystyle a^{\perp}\vee b^{\perp}=(a\wedge b)^{\perp}.$ (2)

    To derive the first equation, first note $ a\le a\vee b$. Then $ (a\vee b)^{\perp}\le a^{\perp}$. Similarly, $ (a\vee b)^{\perp}\le b^{\perp}$. So $ (a\vee b)^{\perp}\le a^{\perp}\wedge b^{\perp}$. For the other inequality, we start with $ a^{\perp}\wedge b^{\perp}\le a^{\perp}$. Then $ a\le (a^{\perp}\wedge b^{\perp})^{\perp}$. Similarly, $ b\le (a^{\perp}\wedge b^{\perp})^{\perp}$. Therefore, $ a\vee b\le (a^{\perp}\wedge b^{\perp})^{\perp}$, which implies that $ a^{\perp}\wedge b^{\perp}\le (a\vee b)^{\perp}$.

  • Conversely, any of two equations in the previous remark can replace the third condition in the definition above. For example, suppose we have the second equation $ a^{\perp}\vee b^{\perp}=(a\wedge b)^{\perp}$. If $ a\le b$, then $ a=a\wedge b$, so $ a^{\perp}=(a\wedge b)^{\perp}=a^{\perp}\vee b^{\perp}$, which shows that $ b^{\perp}\le a^{\perp}$.
  • From the example above, one sees that orthocomplementation need not be unique. An orthocomplemented lattice with a unique orthocomplementation is said to be uniquely orthocomplemented. A uniquely complemented lattice that is also orthocomplemented is uniquely orthocomplemented.
  • Orthocomplementation can be more generally defined over a bounded poset $ P$ by requiring the orthocomplentation operator $ ^{\perp}$ to satisfy conditions 2 and 3 above, and a weaker version of condition 1: $ a\wedge a^{\perp}$ exists and $ =0$. Since $ ^{\perp}$ is an order reversing bijection on $ P$, $ 0^{\perp}=1$ and $ 1^{\perp}=0$. From this, one deduces that $ a\wedge a^{\perp}=0$ iff $ a\vee a^{\perp}=1$. A bounded poset in which an orthocomplementation is defined is called an orthocomplemented poset.
  • In the category of orthocomplemented lattices, the morphism between a pair of objects is a $ \lbrace 0,1\rbrace$-lattice homomorphism $ f$ that preserves orthocomplementation:
    $\displaystyle f(a^{\perp})=f(a)^{\perp}.$

Bibliography

1
G. Birkhoff, Lattice Theory, AMS Colloquium Publications, Vol. XXV, 3rd Ed. (1967).



"orthocomplemented lattice" is owned by CWoo.
(view preamble | get metadata)

View style:

See Also: complemented lattice, orthomodular lattice

Other names:  ortholattice, uniquely orthocomplemented
Also defines:  orthocomplement, orthocomplemented, orthocomplementation, orthocomplemented poset, uniquely orthocomplemented lattice

Attachments:
commutativity relation in an orthocomplemented lattice (Definition) by CWoo
Log in to rate this entry.
(view current ratings)

Cross-references: preserves, objects, morphism, category, iff, order reversing, operator, bounded poset, uniquely complemented lattice, inequality, equation, de Morgan's laws, satisfy, lattice isomorphism, operations, join, meet, onto, one-to-one, bijection, even, cardinality, right, finite, Hasse diagrams, vector subspaces, lattice, implies, image, function, vector space, subspace, complementary subspace, complement, complemented lattice
There are 4 references to this entry.

This is version 17 of orthocomplemented lattice, born on 2006-04-11, modified 2007-05-24.
Object id is 7822, canonical name is OrthocomplementedLattice.
Accessed 4804 times total.

Classification:
AMS MSC: 06C15 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented lattices, orthocomplemented lattices and posets)
 03G12 (Mathematical logic and foundations :: Algebraic logic :: Quantum logic)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)