# Differences

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+
+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}:  $f(x)\wedge y=0\iff x\wedge f(y)=0$
+
+Remark: This is a template.
+
+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]]
+)]
+
+