Monounary Algebras

Abbreviation: MonoUn

Definition

A monounary algebra is a structure $\mathbf{A}=\langle A,f\rangle$ of type $\langle 1\rangle$ such that $f$ is a unary operation on $A$.

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be monounary algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(f(x))=f(h(x))$.

Examples

Example 1: The free unary algebra on one generator is isomorphic to the natural numbers $\mathbb N$. The number 0 is the generator $x$, and the operation $f$ is the successor function, i.e., $f(n)=n+1$.

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

Basic results

Monounary algebras are equivalent to directed graphs in which every vertex has exactly one outgoing edge.

One-generated monounary algebras are either isomorphic to the free one-generated algebra or they are finite and contain a path of length $l$ from the generator to a cycle of length $k$ (where $l\geq 0$ and $k\geq 1$).

Properties

Finite members

Subclasses

Idempotent monounary algebras subvariety

The variety of monounary algebras has countably many subvarieties, each determined by an equation of the form $f^m(x)=f^n(x)$.

Let $j>k\ge 0$ and $m>n\ge 0$. Then $\text{Mod}(f^j(x)=f^k(x)\subseteq\text{Mod}(f^m(x)=f^n(x)$ if and only if $k\le n$ and $(j-k)|(m-n)$.

Hence the lattice of nontrivial subvarieties of monounary algebras is isomorphic to $(\mathbb N,\le)\times (\mathbb N,|)$, which is itself isomorphic to the lattice of divisibility of the natural numbers. The variety $\text{Mod}(x=y)$ of trivial subvarieties is the unique element below the variety $\text{Mod}(f(x)=x)$ (which is term-equivalent to the variety of sets).

Superclasses

Duo-unary algebras subreduct

References