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
Trace: » bilattices