orthocomplemented lattice

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 ${}^{⟂}:L\to M$, called an orthocomplementation, whose image is written $a^{⟂}$ for any $a\in L$, such that

1. $a^{⟂}\in M_a$,

2. ${(a^{⟂})}^{⟂}=a$, and

3. $⟂$ is order-reversing; that is, for any $a,b\in L$, $a\le b$ implies $b^{⟂}\le a^{⟂}$.

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

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,

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 $⟂$ is a bijection. It is one-to-one: if $a^{⟂}=b^{⟂}$, then $a={(a^{⟂})}^{⟂}={(b^{⟂})}^{⟂}=b$. And it is onto: if we pick $a\in M\subseteq L$, then ${(a^{⟂})}^{⟂}=a$. As a result, $M=L$, every element of $L$ is an orthocomplement. Furthermore, we have $0^{⟂}=1$ and $1^{⟂}=0$.

• Let $L^{\prime }$ be the dual lattice of $L$ (a lattice having the same underlying set, but with meet and join operations switched). Then any orthocomplementation $⟂$ 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

  $a^{⟂}\wedge b^{⟂}={(a\vee b)}^{⟂},$ (1)

  $a^{⟂}\vee b^{⟂}={(a\wedge b)}^{⟂}.$ (2)

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

• 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 $a^{⟂}\vee b^{⟂}={(a\wedge b)}^{⟂}$. If $a\le b$, then $a=a\wedge b$, so $a^{⟂}={(a\wedge b)}^{⟂}=a^{⟂}\vee b^{⟂}$, which shows that $b^{⟂}\le a^{⟂}$.

• 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 ${}^{⟂}$ to satisfy conditions 2 and 3 above, and a weaker version of condition 1: $a\wedge a^{⟂}$ exists and $=0$. Since ${}^{⟂}$ is an order reversing bijection on $P$, $0^{⟂}=1$ and $1^{⟂}=0$. From this, one deduces that $a\wedge a^{⟂}=0$ iff $a\vee a^{⟂}=1$. 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 $\{0,1\}$-lattice homomorphism (http://planetmath.org/LatticeHomomorphism) $f$ that preserves orthocomplementation:

  $$f(a^{⟂})=f{(a)}^{⟂}.$$

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