=====Monadic algebras=====

Abbreviation: **MonA**

====Definition====
A \emph{monadic algebra} is a structure $\mathbf{A}=\langle A, \vee, 0, \wedge, 1, \neg, f\rangle$ of type $\langle 2, 0, 2, 0, 1, 1\rangle$ such that

$\langle A, \vee, 0, \wedge, 1, \neg\rangle$ is a [[Boolean algebra]]

$f$ is a \emph{unary closure operator}:  $f(x\vee y)=f(x)\vee f(y)$, $f(0)=0$, $x\le f(x)=f(f(x))$

$f$ is \emph{self conjugated}:  $f(x)\wedge y=0\iff x\wedge f(y)=0$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be monodic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 
$h(x \vee y)=h(x) \vee h(y)$,
$h(\neg x)=\neg h(x)$, 
$h(f(x))=f(h(x))$.

====Definition====
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$

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

====Basic results====


====Properties====
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

^[[Classtype]]                        |variety  |
^[[Equational theory]]                |decidable |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          |yes |
^[[Congruence modular]]               |yes |
^[[Congruence $n$-permutable]]        |yes, $n=2$ |
^[[Congruence regular]]               |yes |
^[[Congruence uniform]]               |yes |
^[[Congruence extension property]]    |yes |
^[[Definable principal congruences]]  |yes |
^[[Equationally def. pr. cong.]]      |yes |
^[[Amalgamation property]]            |yes |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

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


====Subclasses====
  [[...]] subvariety

  [[...]] expansion


====Superclasses====
  [[Boolean algebras]] reduced type

  [[Closure algebras]]


====References====

[(Lastname19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]