F-algebra

In mathematics, specifically in category theory, F-algebras generalize algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature.

F-algebras can also be used to represent data structures used in programming, such as lists and trees.

The main related concepts are initial F-algebras which may serve to encapsulate the induction principle, and the dual construction F-coalgebras.

If C is a category, and F: C → C is an endofunctor of C, then an F-algebra is a tuple (A, α), where A is an object of C and α is a C-morphism F(A) → A. The object A is called carrier of the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple.

A homomorphism from an F-algebra (A, α) to an F-algebra (B, β) is a C-morphism f: A→ B such that f o α = β o F(f), according to the following diagram:

Equipped with these morphisms, F-algebras constitute a category.

The dual construction are F-coalgebras, which are objects A^* together with a morphism α^* : A^* → F(A^*).

Classically, a group is a set G with a binary operation m : G × G → G, satisfying three axioms:

Note that the axiom of closure is included in the symbolic definition of m.

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