### Table of Contents

## Pseudocomplemented distributive lattices

Abbreviation: **pcDLat**

### Definition

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 $x$: $y\leq x^* \iff x\wedge y=0$

##### Morphisms

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:

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

### Definition

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: $0\leq x$

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

$x\wedge 0^*=x$

$0^{**}=0$

### Examples

Example 1:

### Basic results

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

### Properties

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &1

f(3)= &1

f(4)= &

f(5)= &

f(6)= &

f(7)= &

\end{array}$