Pseudocomplemented distributive lattices

Abbreviation: pcDLat

Definition

A 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 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 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:

Basic results

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

Properties

Classtype variety decidable yes yes yes

Finite members

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