Orthomodular lattices

Abbreviation: OMLat

Definition

An orthomodular lattice is an ortholattice $\mathbf{L}=\langle L,\vee,0,\wedge,1,'\rangle$ such that

the orthomodular law holds: $x\le y \implies x\vee(x'\wedge y)=y$.

This law is equivalent to satisfying the identity $x\vee(x'\wedge (x\vee y))=x\vee y$.

Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be orthomodular lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x')=h(x)'$

Examples

Example 1: The closed subspaces of (countably dimensional) Hilbert Space form an orthomodular lattice that is not modular (for finite dimensional vector spaces all subspaces are closed, hence the lattice of closed subspaces is modular).

Example 2: The smallest nonmodular orthomodular lattice has 10 elements and is isomorphic to a parallel sum of a 4-element Boolean algebra and an 8-element Boolean algebra. A failure of the modular law $x\vee(y\wedge(x\vee z))=(x\vee y)\wedge(x\vee z)$ occurs when $x$, $z$ are atoms of the 8-element algebra and $y$ is an atom of the 4-element algebra.

Properties

Classtype variety undecidable no unbounded yes yes no no no no no

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &0\\ f(4)= &1\\ f(5)= &0\\ f(6)= &1\\ f(7)= &0\\ f(8)= &2\\ \end{array}$

Many Greechie diagrams of orthomodular lattices with blocks containing 3 atoms have been computed at http://cs.anu.edu.au/~Brendan.McKay/nauty/greechie.html