orthocomplemented lattice

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

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 inherited from L. For each aL, let MaM be the set of complements of a. L is said to be orthocomplemented if there is a function :LM, called an orthocomplementation, whose image is written a for any aL, such that

  1. 1.


  2. 2.

    (a)=a, and

  3. 3.

    is order-reversing; that is, for any a,bL, ab implies ba.

The element a is called an orthocomplement of a (via ).

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

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


  • From the first condition above, we see that an orthocomplementation is a bijection. It is one-to-one: if a=b, then a=(a)=(b)=b. And it is onto: if we pick aML, then (a)=a. As a result, M=L, every element of L is an orthocomplement. Furthermore, we have 0=1 and 1=0.

  • Let L be the dual lattice of L (a lattice having the same underlying set, but with meet and join operationsMathworldPlanetmath switched). Then any orthocomplementation can be viewed as a lattice isomorphismMathworldPlanetmath between L and L.

  • From the above conditions, it follows that elements of L satisfy the de Morgan’s laws: for a,bL, we have

    ab=(ab), (1)
    ab=(ab). (2)

    To derive the first equation, first note aab. Then (ab)a. Similarly, (ab)b. So (ab)ab. For the other inequality, we start with aba. Then a(ab). Similarly, b(ab). Therefore, ab(ab), which implies that ab(ab).

  • 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 ab=(ab). If ab, then a=ab, so a=(ab)=ab, which shows that ba.

  • 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 latticeMathworldPlanetmath that is also orthocomplemented is uniquely orthocomplemented.

  • Orthocomplementation can be more generally defined over a bounded poset P by requiring the orthocomplentation operator to satisfy conditions 2 and 3 above, and a weaker version of condition 1: aa exists and =0. Since is an order reversing bijection on P, 0=1 and 1=0. From this, one deduces that aa=0 iff aa=1. A bounded poset in which an orthocomplementation is defined is called an orthocomplemented poset.

  • In the categoryMathworldPlanetmath of orthocomplemented lattices, the morphismMathworldPlanetmath between a pair of objects is a {0,1}-lattice homomorphism (http://planetmath.org/LatticeHomomorphism) f that preserves orthocomplementation:



  • 1 G. Birkhoff, Lattice Theory, AMS Colloquium Publications, Vol. XXV, 3rd Ed. (1967).
Title orthocomplemented lattice
Canonical name OrthocomplementedLattice
Date of creation 2013-03-22 15:50:36
Last modified on 2013-03-22 15:50:36
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 20
Author CWoo (3771)
Entry type Definition
Classification msc 03G12
Classification msc 06C15
Synonym ortholattice
Synonym uniquely orthocomplemented
Related topic ComplementedLattice
Related topic OrthomodularLattice
Defines orthocomplement
Defines orthocomplemented
Defines orthocomplementation
Defines orthocomplemented poset
Defines uniquely orthocomplemented lattice