Orthocomplemented lattice

From Wikipedia, the free encyclopedia

An orthocomplemented lattice is an algebraic structure consisting of a bounded lattice $<L, \vee, \wedge, 0, 1 >$ along with a unary function $\perp : L \rightarrow L$ called an orthocomplementation. This function must satisfy for any $a,b \in L$

Complement law: $a^ \perp \vee a = 1$ and $a^ \perp \wedge a = 0$

Involution law: $a^{\perp \perp} = a$

Order-reversing law: if $a \leq b$ then $b^\perp \leq a^\perp$

For a given element $a$ the element $a^\perp$ is called the orthocomplement of $a$ . From the axioms it can be shown that this element is unique.

Orthocomplemented lattices satisfy de Morgan's laws:

$(a\vee b)^\perp = a^\perp \wedge b^\perp$
$(a\wedge b)^\perp = a^\perp \vee b^\perp$

Boolean algebras are a special case of orthocomplemented lattices, which in turn are special cases of complemented lattices. These structures are most often used in quantum logic, where the closed subspaces of a separable Hilbert space represent quantum propositions and behave as an orthocomplemented lattice.