## 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.

### Properties

Classtype variety undecidable if $|I|>2$ no no no no no no no no no

### Finite members

Depends on $I$

### Superclasses

Duo-unary algebras subreduct