−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″=x and that (x∨y)′=x′∧y′.
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:A→B that is a homomorphism: h(x∨y)=h(x)∨h(y), h(x⋅y)=h(x)⋅h(y), h(x′)=h(x)′ and h(1)=1.
Examples
Example 1:
Basic results
Properties
Finite members
$\begin{array}{lr}
f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\
\end{array}\begin{array}{lr}
f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\
\end{array}$