Abbreviation: DDblpAlg

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

$\langle L,\vee,0,\wedge,1,^*\rangle$ is a distributive p-algebras

$\langle L,\vee,0,\wedge,1,^+\rangle$ is a distributive dual p-algebras

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

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

Example 1:

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

Double Stone algebras

Distributive p-algebras

Distributive dual p-algebras