unary_algebras - MathStructures

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===== Unary Algebras =====

Abbreviation: **Unar**

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

^[[Classtype]]                        |variety |
^[[Equational theory]]                |undecidable if $|I|>2$ |
^[[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====

Depends on $I$

====Subclasses====

[[Permutation unary algebras]] subvariety

====Superclasses====

[[Duo-unary algebras]] subreduct


====References====

Trace: • horn_formulas • quasigroups • abelian_p-groups • first-order_theory • clifford_semigroups • bounded_distributive_lattices • continuous_lattices • dunn_monoids • congruence_distributive • inverse_semigroups