Abbreviation: ImpLat
An \emph{implicative lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\to\rangle$ such that
$\langle A,\vee,\wedge\rangle$ is a distributive lattices $\to$ is an implication:
$x\to(y\vee z) = (x\to y)\vee(x\to z)$
$x\to(y\wedge z) = (x\to y)\wedge(x\to z)$
$(x\vee y)\to z = (x\to z)\wedge(y\to z)$
$(x\wedge y)\to z = (x\to z)\vee(y\to z)$
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:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\vee y)=h(x)\wedge h(y)$, $h(x\to y)=h(x)\to h(y)$
Nestor G. Martinez,H. A. Priestley,\emph{On Priestley spaces of lattice-ordered algebraic structures}, Order, \textbf{15}1998,297–323MRreview
Nestor G. Martinez,\emph{A simplified duality for implicative lattices and $l$-groups}, Studia Logica, \textbf{56}1996,185–204MRreview
Example 1:
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &
f(5)= &
f(6)= &
f(7)= &
f(8)= &
f(9)= &
f(10)= &
\end{array}$