Fano resonance

In mathematics, specifically in category theory, an F-algebra is a structure defined according to a (endo-)functor F. F-algebras can be used to represent data structures used in programming, such as lists and trees. Initial F-algebras encapsulate an induction principle.

F-algebras are dual to F-coalgebras.

The commutative diagram, which defines a property required by morphisms of the original category, so that they can be morphism of the newly defined category of F-algebras.

Definition

If ${\mathcal {C}}$ is a category, and

$F:{\mathcal {C}}\longrightarrow {\mathcal {C}}$

is an endofunctor of ${\mathcal {C}}$, then an F-algebra is an object $A$ of ${\mathcal {C}}$ together with a ${\mathcal {C}}$-morphism

$\alpha :F(A)\longrightarrow A$.

In this sense F-algebras are dual to F-coalgebras.

A homomorphism from an F-algebra $(A,\alpha )$ to an F-algebra $(B,\beta )$ is a ${\mathcal {C}}$-morphism

$f:A\longrightarrow B$

such that

$f\circ \alpha =\beta \circ F(f)$,

see picture.

Thus the F-algebras constitute a category.

Example

Consider the functor $F:\mathbf {Set} \to \mathbf {Set}$ that sends a set $X$ to $1+X$. Here, Set denotes the category of sets, $+$ denotes the usual coproduct given by disjoint union, and 1 is a terminal object (i.e. any singleton set). Then the set N of natural numbers together with the function $[\mathrm {zero} ,\mathrm {succ} ]:1+\mathbb {N} \to \mathbb {N}$, which is the coproduct of the functions $\mathrm {zero} :1\to \mathbb {N}$ (whose image is 0) and $\mathrm {succ} :\mathbb {N} \to \mathbb {N}$ (which sends an integer n to n+1), is an F-algebra.

Initial F-algebra

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If the category of F-algebras for a given endofunctor F has an initial object, it is called an initial algebra. The algebra $(\mathbb {N} ,[\mathrm {zero} ,\mathrm {succ} ])$ in the above example is an initial algebra. Various finite data structures used in programming, such as lists and trees, can be obtained as initial algebras of specific endofunctors.

Types defined by using least fixed point construct with functor F can be regarded as an initial F-algebra, provided that parametricity holds for the type.

See also Universal algebra.

Terminal F-coalgebra

In a dual way, similar relationship exists between notions of greatest fixed point and terminal F-coalgebra, these can be used for allowing potentially infinite objects while maintaining strong normalization property. In the strongly normalizing Charity programming language (i.e. each program terminates in it), coinductive data types can be used achieving surprising results, e.g. defining lookup constructs to implement such “strong” functions like the Ackermann function.

See also

• Algebraic data type
• Catamorphism

Further reading

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  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Categorical programming with inductive and coinductive types by Varmo Vene
• Philip Wadler: Recursive types for free! University of Glasgow, June 1990. Draft.
• Algebra and coalgebra from CLiki
• B. Jacobs, J.Rutten: A Tutorial on (Co) Algebras and (Co) Induction. Bulletin of the European Association for Theoretical Computer Science, vol. 62, 1997