monounary_algebras - MathStructures

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===== Monounary Algebras =====

Abbreviation: **MonoUn**

====Definition====
A \emph{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====

^[[Classtype]]                        |variety |
^[[Equational theory]]                |decidable |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   |no |
^[[Residual size]]                    | |
^[[Congruence distributive]]          |no |
^[[Congruence modular]]               |no |
^[[Congruence $n$-permutable]]        |no |
^[[Congruence regular]]               |no |
^[[Congruence uniform]]               |no |
^[[Congruence extension property]]    |no |
^[[Definable principal congruences]]  |no |
^[[Equationally def. pr. cong.]]      |no |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====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====

Trace: • wajsberg_hoops • nonassociative_relation_algebras • hilbert_algebras • sqrt-quasi-mv-algebras • brouwerian_algebras • bck-algebras • dedekind_domains • modal_algebras • torsion_groups • neardistributive_lattices