Abbreviation: BMon
A \emph{Boolean monoid} is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\cdot,e\rangle$ such that
$\langle A,\vee,0, \wedge,1,\neg\rangle $ is a Boolean algebra
$\langle A,\cdot,e\rangle $ is a monoids
$\cdot$ is \emph{join-preserving} in each argument: $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z) \mbox{ and } x\cdot (y\vee z)=(x\cdot y)\vee (x\cdot z)$
$\cdot$ is \emph{normal} in each argument: $0\cdot x=0 \mbox{ and } x\cdot 0=0$
Remark:
Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$, $e$:
$h(x\cdot y)=h(x)\cdot h(y) \mbox{ and } h(e)=e$
Example 1:
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &0
f(4)= &9
f(5)= &0
f(6)= &0
f(7)= &0
f(8)= &258
\end{array}$