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## 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 $\oplus$, $\otimes$: $\neg(x\vee y)=\oplus x\wedge\oplus y$, $\neg(x\otimes y)=\neg x\otimes\neg y$.

##### Morphisms

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

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

### 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)= &

### Subclasses

### Superclasses

### References

Trace: » bilattices