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bilattices [2012/06/16 00:15]
jipsen created
bilattices [2012/06/16 00:18] (current)
jipsen
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====Definition==== ====Definition====
-A \emph{bilattice} is a structure $\mathbf{L}=\langle L,\vee,\wedge,\oplus,\otimes,\neg,rangle$ such that+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,\vee,\wedge\rangle $ is a [[lattice]],
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$\neg$ is a De Morgan operation for $\vee$, $\wedge$: $\neg(x\vee y)=\neg x\wedge\neg y$, $\neg\neg x=x$ and $\neg$ 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 $\oplus$, $\otimes$: $\neg(x\vee y)=\oplus x\wedge\oplus y$, $\neg(x\otimes y)=\neg x\otimes\neg y$.+$\neg$ commutes with $\oplus$, $\otimes$: $\neg(x\oplus y)=\neg x\oplus\neg y$, $\neg(x\otimes y)=\neg x\otimes\neg y$.
==Morphisms== ==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+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: homomorphism:
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f(9)= &\\ f(9)= &\\
f(10)= &\\ f(10)= &\\
 +\end{array}$