Abbreviation: MonA

### Definition

A 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 unary closure operator: $f(x\vee y)=f(x)\vee f(y)$, $f(0)=0$, $x\le f(x)=f(f(x))$

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

$...$ is …: $axiom$

$...$ is …: $axiom$

Example 1:

### 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 decidable yes yes yes, $n=2$ yes yes yes yes yes yes

### 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]]