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## Unary Algebras

Abbreviation: **Unar**

### Definition

A ** unary algebra** is a structure $\mathbf{A}=\langle A,(f_i:\in I)\rangle$ of type $\langle 1: i\in I\rangle$ such that
$f_i$ is a unary operation on $A$ for all $i\in I$.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be unary algebras over the same index set $I$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(f_i(x))=f_i(h(x))$ for all $i\in I$.

### Examples

Example 1: The free unary algebra on one generator is isomorphic to $I^*$, the set of all $n$-tuples of $I$ for $n\in\omega$. The empty tuple is the generator $x$, and the operations $f_i$ are defined by $f_i((i_1,\ldots,i_n))=(i,i_1,\ldots,i_n)$.

The free unary algebra on $X$ generators is a union of $|X|$ disjoint copies of the one-generated free algebra.

### Basic results

### Properties

### Finite members

Depends on $I$

### Subclasses

[[Permutation unary algebras]] subvariety

### Superclasses

[[Duo-unary algebras]] subreduct

### References

Trace: » unary_algebras