Abbreviation: MA

A \emph{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 \emph{join-preserving}: $\diamond(x\vee y)=\diamond x\vee \diamond y$

$\diamond$ is \emph{normal}: $\diamond 0=0$

Remark: Modal algebras provide algebraic models for modal logic. The operator $\diamond$ is the \emph{possibility operator}, and the \emph{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