de_morgan_algebras - MathStructures

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=====De Morgan algebras=====

Abbreviation: **DeMA**
====Definition====
A \emph{De Morgan algebra} is a structure $\mathbf{A}=\langle A,\vee
,0,\wedge ,1,\neg\rangle $ such that


$\langle A,\vee ,0,\wedge ,1\rangle $ is a bounded distributive
lattice


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


Remark: 
It follows that $\neg ( x\vee y) =\neg x\wedge \neg y$, $\ \neg 1=0$ and $\neg 0=1$ (e.g. $\neg 1=\neg 1\vee 0=\neg 1\vee\neg\neg 0=
\neg(1\wedge\neg 0)=\neg\neg 0=0$). Thus $\neg$ is a dual automorphism.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be De Morgan algebras. 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: Let $\{0<a,b<1\}$ be the 4-element lattice with $a,b$ incomparable, and define $'$ by $0'=1,a'=a,b'=b$.
====Basic results====

The algebra in Example 1 generates the variety of De Morgan algebras, see e.g. http://www.math.uic.edu/~kauffman/DeMorgan.pdf

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
^[[Locally finite]]  |yes |
^[[Residual size]]  | 4 |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &3\\
f(5)= &1\\
f(6)= &4\\
f(7)= &2\\
f(8)= &9\\
f(9)= &5\\
f(10)= &14\\
\end{array}$

====Subclasses====
[[Kleene logic algebras]] 

====Superclasses====
[[Ockham algebras]] 


====References====

[(Ln19xx>
)]

Trace: • meet-semidistributive_lattices • algebraic_semilattices • sqrt-quasi-mv-algebras • pseudo_mv-algebras • commutative_idempotent_integral_involutive_fl-algebras • peirce_algebras • near-rings • involutive_residuated_lattices • heyting_algebras • cylindric_algebras