Involutive lattices

Abbreviation: InvLat

Definition

An 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.

Morphisms

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:

Properties

Classtype variety no unbounded yes yes

Finite members

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