## De Morgan monoids

Abbreviation: DMMon

### Definition

A De Morgan monoid is a structure $\mathbf{A}=\langle A,\vee,\wedge,\cdot,1,',\rangle$ of type $\langle 2,2,2,0,1\rangle$ such that

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

$\langle A,\cdot,1\rangle$ is a commutative monoid,

$\cdot$ is involutive residuated: $x\cdot y\le z\iff y\le (z'\cdot x)'$ and

$\cdot$ is square-increasing: $x\le x\cdot x$.

Remark: It follows that $x''=x$ 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 $h(1)=1$.

Example 1:

### Properties

Classtype variety

### Finite members

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

... subvariety

... expansion

... supervariety

... subreduct