Abbreviation: OLat

An \emph{ortholattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,1,'\rangle$ such that

$\langle L,\vee,0,\wedge,1\rangle$ is a bounded lattice

$'$ is complementation: $x\vee x'=1$, $x\wedge x'=0$, $x''=x$

$'$ satisfies De Morgan's laws: $(x\vee y)'=x'\wedge y'$, $(x\wedge y)'=x'\vee y'$

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

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

Example 1: $\langle P(S),\cup ,\emptyset ,\cap ,S\rangle$, the collection of subsets of a set $S$, with union, empty set, intersection, and the whole set $S$.

Orthomodular lattices

Complemented lattices