Abbreviation: MA

A modal algebra is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\diamond\rangle$ such that

$\langle A,\vee,0, \wedge,1,\neg\rangle $ is a Boolean algebras

$\diamond$ is join-preserving: $\diamond(x\vee y)=\diamond x\vee \diamond y$

$\diamond$ is normal: $\diamond 0=0$

Remark: Modal algebras provide algebraic models for modal logic. The operator $\diamond$ is the possibility operator, and the necessity operator $\Box$ is defined as $\Box x=\neg\diamond\neg x$.

Let $\mathbf{A}$ and $\mathbf{B}$ be modal algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:

$h(\diamond x)=\diamond h(x)$

Example 1:

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

Closure algebras

Boolean algebras with operators