Bilattices

Abbreviation: Bilat

Definition

A bilattice is a structure $\mathbf{L}=\langle L,\vee,\wedge,\oplus,\otimes,\neg\rangle$ such that

$\langle L,\vee,\wedge\rangle $ is a lattice,

$\langle L,\oplus,\otimes\rangle $ is a lattice,

$\neg$ is a De Morgan operation for $\vee$, $\wedge$: $\neg(x\vee y)=\neg x\wedge\neg y$, $\neg\neg x=x$ and

$\neg$ commutes with $\oplus$, $\otimes$: $\neg(x\oplus y)=\neg x\oplus\neg y$, $\neg(x\otimes y)=\neg x\otimes\neg y$.

Morphisms

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\oplus y)=h(x)\oplus h(y)$, $h(x\otimes y)=h(x)\otimes h(y)$, $h(\neg x)=\neg h(x)$

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &0\\ f(3)= &0\\ f(4)= &1\\ f(5)= &3\\ f(6)= &32\\ f(7)= &284\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

Superclasses

References