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monadic_algebras [2010/07/29 15:46] (current)
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 +=====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}:  $f(x)\wedge y=0\iff x\wedge f(y)=0$
 +
 +Remark: This is a template.
 +If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page.
 +
 +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]]
 +)]
 +
 +