F-Algebra

F- algebra is a structure which is based solely on Funktoreigenschaften.

Dual to the notion of F -algebra of F- coalgebra

Definition

It should be a category and a functor. Each morphism is then an algebra. The object is called a carrier of.

Homomorphisms

Are and algebras in so called a morphism according to the property of homomorphism.

Initial F - algebras

The homomorphisms between algebras for a fixed functor in turn form another category in which the objects are algebras. An initial object of this category is called initial algebra. If initial, as - algebra isomorphic to how the diagram

Shows. It is the only homomorphism from to. Therefore, the left square commutes. The right commutes trivially. Thus, the outer rectangle commutes and is an algebra homomorphism from to. But there is initial, must be. On the other hand, because of the left rectangle and the equation just found.

The importance of initial algebras lies in the fact that certain recursive structures can be mapped in an orderly manner. Indeed, if an initial algebra, and any other - algebra, then there exists and there is exactly one morphism, the solution of the equation. This means Katamorphismus.

Existence theorems for initial algebras

• In SetC, the category of countable sets and functions, is available for every Endofunktor an initial algebra.
• In RELC, the category of countable sets and relations, exist at any Endofunktor an initial algebra.