orthomodular lattice
Orthogonality Relations
Let $L$ be an orthocomplemented lattice and $a,b\in L$. $a$ is said to be orthogonal^{} to $b$ if $a\le {b}^{\u27c2}$, denoted by $a\u27c2b$. If $a\le {b}^{\u27c2}$, then $b={b}^{\u27c2\u27c2}\le {a}^{\u27c2}$, so $\u27c2$ is a symmetric relation^{} on $L$. It is easy to see that, for any $a,b\in L$, $a\u27c2b$ implies $a\wedge b=0$, and $a\u27c2{a}^{\u27c2}$.
For any $a\in L$, define $M(a):=\{c\in L\mid c\u27c2a\text{and}1=c\vee a\}$. An element of $M(a)$ is called an orthogonal complement^{} of $a$. We have ${a}^{\u27c2}\in M(a)$, and any orthogonal complement of $a$ is a complement^{} of $a$.
If we replace the $1$ in $M(a)$ by an arbitrary element $b\ge a$, then we have the set
$$M(a,b):=\{c\in L\mid c\u27c2a\text{and}b=c\vee a\}.$$ 
An element of $M(a,b)$ is called an orthogonal complement of $a$ relative to $b$. Clearly, $M(a)=M(a,1)$. Also, for $a,c\le b$, $c\in M(a,b)$ iff $a\in M(c,b)$. As a result, we can define a symmetric^{} binary operator $\oplus $ on $[0,b]$, given by $b=a\oplus c$ iff $c\in M(a,b)$. Note that $b=b\oplus 0$.
Before the main definition, we define one more operation^{}: $ba:=b\wedge {a}^{\u27c2}$. Some properties: (1) $aa=0$, $a0=a$, $0a=0$, $a1=0$, and $1a={a}^{\u27c2}$; (2) $ba={a}^{\u27c2}{b}^{\u27c2}$; and (3) if $a\le b$, then $a\u27c2(ba)$ and $a\oplus (ba)\le b$.
Definition
A lattice^{} $L$ is called an orthomodular lattice if

1.
$L$ is orthocomplemented, and

2.
(orthomodular law) if $x\le y$, then $y=x\oplus (yx)$.
The orthomodular law can be restated as follows: if $x\le y$, then $y=x\vee (y\wedge {x}^{\u27c2})$. Equivalently, $x\le y$ implies $y=(y\wedge x)\vee (y\wedge {x}^{\u27c2})$. Note that the equation is automatically true in an arbitrary distributive lattice^{}, even without the assumption^{} that $x\le y$.
For example, the lattice $\u2102(H)$ of closed subspaces of a hilbert space $H$ is orthomodular. $\u2102(H)$ is modular iff $H$ is finite dimensional. In addition, if we give the set $\mathbb{P}(H)$ of (bounded) projection operators on $H$ an ordering structure^{} by defining $P\le Q$ iff $P(H)\le Q(H)$, then $\mathbb{P}(H)$ is lattice isomorphic to $\u2102(H)$, and hence orthomodular.
A simple example of an orthocomplemented lattice that is not orthomodular is the benzene:
$$\text{xymatrix}\mathrm{\&}1\text{ar}\mathrm{@}[ld]\text{ar}\mathrm{@}[rd]\mathrm{\&}b\text{ar}\mathrm{@}[d]\mathrm{\&}\mathrm{\&}{a}^{\u27c2}\text{ar}\mathrm{@}[d]a\text{ar}\mathrm{@}[rd]\mathrm{\&}\mathrm{\&}{b}^{\u27c2}\text{ar}\mathrm{@}[ld]\mathrm{\&}0\mathrm{\&}$$ 
Note that $a\le b$, but $a\vee (b\wedge {a}^{\u27c2})=a\vee 0=a\ne b$.
An nice example of an orthomodular lattice that is not modular can be found in the reference below.
Remarks.

•
Orthomodular lattices were first studied by John von Neumann and Garett Birkhoff, when they were trying to develop the logic of quantum mechanics (http://planetmath.org/QuantumLogic) by studying the structure of the lattice $\mathbb{P}(H)$ of projection operators on a Hilbert space $H$. However, the term was coined by Irving Kaplansky, when it was realized that $\mathbb{P}(H)$, while orthocomplemented, is not modular. Rather, it satisfies a variant of the modular law as indicated above.

•
More generally, an orthomodular poset $P$ is an orthocomplemented poset such that

(a)
given any pair of orthogonal elements $x,y\in P$ ($x\le {y}^{\u27c2}$), their greatest lower bound^{} exists ($x\vee y$ exists). Simply put, $x\u27c2y$ implies $x\vee y\in P$.

(b)
for any $x,y\in P$ such that $x\le y$, the orthomodular law holds (the right hand side of the orthomodular law exists via the first condition).
From this definition, we see that an orthomodular lattice is just an orthomodular poset that is also a lattice.

(a)
References
 1 L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
Title  orthomodular lattice 
Canonical name  OrthomodularLattice 
Date of creation  20130322 16:33:06 
Last modified on  20130322 16:33:06 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06C15 
Classification  msc 81P10 
Classification  msc 03G12 
Related topic  OrthocomplementedLattice 
Related topic  LatticeOfProjections 
Defines  orthomodular poset 
Defines  orthogonal 
Defines  orthogonal complement 
Defines  relative orthogonal complement 