## Boolean monoids

Abbreviation: BMon

### Definition

A 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 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 normal in each argument: $0\cdot x=0 \mbox{ and } x\cdot 0=0$

Remark:

##### Morphisms

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:

### Properties

Classtype variety no unbounded yes yes yes, $n=2$ yes yes yes no no

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &0\\ f(4)= &9\\ f(5)= &0\\ f(6)= &0\\ f(7)= &0\\ f(8)= &258\\ \end{array}$