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De Morgan algebras

Abbreviation: DeMA

Definition

A 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

Properties

Classtype Variety Yes Yes Yes

Finite members

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