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Unary Algebras
Abbreviation: Unar
Definition
A \emph{unary algebra} is a structure A=⟨A,(fi:∈I)⟩ of type ⟨1:i∈I⟩ such that fi is a unary operation on A for all i∈I.
Morphisms
Let A and B be unary algebras over the same index set I. A morphism from A to B is a function h:A→B that is a homomorphism: h(fi(x))=fi(h(x)) for all i∈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∈ω. The empty tuple is the generator x, and the operations fi are defined by fi((i1,…,in))=(i,i1,…,in).
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