Abbreviation: Bilat

A \emph{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 $\oplus$, $\otimes$: $\neg(x\oplus y)=\neg x\oplus\neg y$, $\neg(x\otimes y)=\neg x\otimes\neg y$.

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:

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

Example 1:

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

Interlaced bilattices

pre-bilattices