Abbreviation: DblStAlg

A double Stone algebra is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1,^*\rangle $ such that

$\langle L,\vee,0,\wedge,1,^*\rangle $ is a Stone algebras

$\langle L,\wedge,1,\vee,0,^*\rangle $ is a Stone algebras

Let $\mathbf{L}$ and $\mathbf{M}$ be double Stone algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$, $h(x^*)=h(x)^*$

Example 1:

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

Boolean algebras

Stone algebras