## Allegories

Abbreviation: All

### Definition

An allegory is an expanded category $\mathbf{M}=\langle M,\circ,\text{dom},\text{rng},\text{id},\vee,\wedge,^\smile\rangle$ such that

$...$ is …: $...$

$...$ is …: $...$

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 allegories. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a functor $F:A\rightarrow B$ that also preserves the new operations: $h(x ... y)=h(x) ... h(y)$

### 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 partial algebras

### 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

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1) %F. Lastname, Title, Journal, 1, 23–45 MRreview