Abbreviation: CdLat

A \emph{complemented lattice} is a bounded lattices $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ such that

every element has a complement: $\exists y(x\vee y=1\mbox{ and }x\wedge y=0)$

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$

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 $S$.

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

Complemented modular lattices

Bounded lattices