Abbreviation: pcDLat

A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that

$\langle L,\vee,0,\wedge\rangle $ is a distributive lattices with bottom element $0$

$x^*$ is the \emph{pseudo complement} of $x$: $y\leq x^* \iff x\wedge y=0$

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

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

A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that

$\langle L,\vee,0,\wedge\rangle $ is a distributive lattices

$0$ is the bottom element: $0\leq x$

$x\wedge(x\wedge y)^*=x\wedge y^*$

$x\wedge 0^*=x$

$0^{**}=0$

Example 1:

Pseudocomplemented distributive lattices are term equivalent to distributive p-algebras.

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

Distributive double p-algebras

Distributive lattices