=====De Morgan monoids=====

Abbreviation: **DMMon**

====Definition====
A \emph{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 $h(1)=1$.


====Examples====
Example 1: 

====Basic results====


====Properties====

^[[Classtype]]                        |variety |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====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====

[(Ln19xx>
)]