−Table of Contents
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: x⋅y≤z⟺y≤(z′⋅x)′ and
⋅ is square-increasing: x≤x⋅x.
Remark: It follows that x″ and that (x\vee y)'=x'\wedge y'.
Note that a De Morgan monoid is the same thing as a commutative distributive involutive residuated lattice.
Morphisms
Let \mathbf{A} and \mathbf{B} be De Morgan monoids. 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(x \cdot y)=h(x) \cdot h(y), h(x')=h(x)' and h(1)=1.
Examples
Example 1:
Basic results
Properties
Finite members
$\begin{array}{lr}
f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\
\end{array} \begin{array}{lr}
f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\
\end{array}$