Abbreviation: InvLat

An \emph{involutive lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\neg\rangle$ such that

$\langle A,\vee,\wedge\rangle$ is a lattices

$\neg$ is a De Morgan involution: $\neg( x\wedge y) =\neg x\vee \neg y$, $\neg\neg x=x$

Remark: It follows that $\neg ( x\vee y) =\neg x\wedge \neg y$. Thus $\neg$ is a dual automorphism.

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

$h(x\vee y)=h(x)\vee h(y)$, $h(\neg x)=\neg h(x)$

Example 1:

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= & f(5)= & f(6)= & f(7)= & f(8)= & f(9)= & f(10)= & \end{array}$

De Morgan algebras

Lattices