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De Morgan monoids

Abbreviation: DMMon

Definition

A \emph{De Morgan monoid} is a structure A=A,,,,1,, of type 2,2,2,0,1 such that

A,, is a distributive lattice,

A,,1 is a commutative monoid,

is involutive residuated: xyzy(zx) and

is square-increasing: xxx.

Remark: It follows that x=x and that (xy)=xy.

Note that a De Morgan monoid is the same thing as a commutative distributive involutive residuated lattice.

Morphisms

Let A and B be De Morgan monoids. A morphism from A to B is a function h:AB that is a homomorphism: h(xy)=h(x)h(y), h(xy)=h(x)h(y), h(x)=h(x) and h(1)=1.

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr}

f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\

\end{array}\begin{array}{lr}

f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Subclasses

... subvariety

... expansion

Superclasses

... supervariety

... subreduct

References


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