bilattices - MathStructures

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=====Bilattices=====

Abbreviation: **Bilat**

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

==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====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  |undecidable |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |

====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====
[[Interlaced bilattices]] 

====Superclasses====
[[pre-bilattices]] 


====References====

[(AB2002>
)]

Trace: • generalized_boolean_algebras • lattice-ordered_semigroups • finite_commutative_semigroups • cancellative_commutative_semigroups

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