involutive_lattices - MathStructures

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=====Involutive lattices=====

Abbreviation: **InvLat**
====Definition====
An \emph{involutive lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\neg\rangle$ such that


$\langle A,\vee,\wedge\rangle$ is a [[lattices]]


$\neg$ is a De Morgan involution:  $\neg( x\wedge
y) =\neg x\vee \neg y$, $\neg\neg x=x$


Remark: 
It follows that $\neg ( x\vee y) =\neg x\wedge \neg y$. Thus $\neg$ is a dual automorphism.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
homomorphism: 

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

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[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]]  | |
====Finite members====

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

====Subclasses====
[[De Morgan algebras]] 

====Superclasses====
[[Lattices]] 


====References====

[(Ln19xx>
)]

Trace: • quasitrivial_groupoids • fle-algebras • lie_algebras • linear_logic_algebras • topological_spaces • rings_with_identity • complete_semilattices • dedekind_domains • algebras • weakly_associative_relation_algebras