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{{DISPLAYTITLE:''F''-algebra}}
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 [[Mathematical programming|programming]], such as [[List (computing)|list]]s and trees. Initial ''F''-algebras encapsulate an induction principle.


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''F''-algebras are dual to [[F-coalgebra|''F''-coalgebra]]s.  


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[[File:Commutative diagram defining F-algebra.png|thumb|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.]]


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==Definition==
If <math>\mathcal{C}</math> is a [[category (mathematics)|category]], and


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:<math>F : \mathcal{C}\longrightarrow \mathcal{C}</math>


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is an [[endofunctor]] of <math>\mathcal{C}</math>, then an '''''F''-algebra''' is an object <math>A</math> of <math>\mathcal{C}</math> together with a <math>\mathcal{C}</math>-[[morphism]]
 
:<math>\alpha : F(A) \longrightarrow A</math>.
 
In this sense ''F''-algebras are dual to ''F''-coalgebras.  
 
A [[homomorphism]] from an ''F''-algebra <math>(A, \alpha)</math> to an ''F''-algebra <math>(B, \beta)</math> is a <math>\mathcal{C}</math>-morphism
 
:<math>f:A\longrightarrow B</math>
 
such that
 
:<math> f\circ \alpha = \beta \circ F(f)</math>,
 
see picture.
 
Thus the ''F''-algebras constitute a category.
 
== Example ==
 
Consider the functor <math>F: \mathbf{Set} \to\mathbf{Set}</math> that sends a set <math>X</math> to <math>1+X</math>. Here, '''Set''' denotes the [[category of sets]], <math>+</math> 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 number]]s together with the function <math>[\mathrm{zero},\mathrm{succ}] : 1+\mathbb{N} \to \mathbb{N}</math>, which is the coproduct of the functions <math>\mathrm{zero} : 1 \to \mathbb{N}</math> (whose image is ''0'') and <math> \mathrm{succ} : \mathbb{N} \to \mathbb{N}</math> (which sends an integer ''n'' to ''n+1''), is an ''F''-algebra.
 
== Initial ''F''-algebra ==
{{main|Initial algebra}}
 
If the category of ''F''-algebras for a given endofunctor ''F'' has an [[initial object]], it is called an '''initial algebra'''. The algebra <math>(\mathbb{N}, [\mathrm{zero},\mathrm{succ}])</math> in the above example is an initial algebra. Various [[finite set|finite]] [[data structures]] used in [[Mathematical programming|programming]], such as [[List (computing)|list]]s 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.<ref name=free-rectypes>Philip Wadler: [http://homepages.inf.ed.ac.uk/wadler/papers/free-rectypes/free-rectypes.txt Recursive types for free!] University of Glasgow, June 1990. Draft.
</ref>
 
See also [[Universal algebra]].
 
== Terminal ''F''-coalgebra ==
 
In a [[Duality (mathematics)|dual]] way, similar relationship exists between notions of [[greatest fixed point]] and terminal ''F''-coalgebra, these can be used for allowing [[Actual infinity|potentially infinite]] objects while maintaining [[Normalization property (lambda-calculus)|strong normalization property]].<ref name=free-rectypes/> In the strongly normalizing [[Charity (programming language)|Charity]] programming language (i.e. each program terminates in it), [[Coinduction|coinductive]] data types can be used achieving surprising results, e.g. defining [[Lookup table|lookup]] constructs to implement such [[Computability theory (computer science)|“strong”]] functions like the [[Ackermann function]].<ref>Robin Cockett: Charitable Thoughts ([ftp://ftp.cpsc.ucalgary.ca/pub/projects/charity/literature/papers_and_reports/charitable.ps ps] and [ftp://ftp.cpsc.ucalgary.ca/pub/projects/charity/literature/papers_and_reports/charitable.ps.gz ps.gz])</ref>
 
== See also ==
* [[Algebraic data type]]
* [[Catamorphism]]
 
== Notes ==
<references/>
 
== Further reading ==
* {{cite book|first=Benjamin C.|last=Pierce|authorlink=Benjamin C. Pierce|title=Basic Category Theory for Computer Scientists|chapter=''F''-Algebras|isbn=0-262-66071-7}}
 
== External links ==
 
* [http://www.cs.ut.ee/~varmo/papers/thesis.pdf Categorical programming with inductive and coinductive types] by Varmo Vene
* Philip Wadler: [http://homepages.inf.ed.ac.uk/wadler/papers/free-rectypes/free-rectypes.txt Recursive types for free!] University of Glasgow, June 1990. Draft.
* [http://tunes.org/wiki/algebra_20and_20coalgebra.html Algebra and coalgebra] from CLiki
* B. Jacobs, J.Rutten: [http://www.cs.ru.nl/~bart/PAPERS/JR.pdf A Tutorial on (Co) Algebras and (Co) Induction. Bulletin of the European Association for Theoretical Computer Science], vol. 62, 1997
 
[[Category:Category theory]]
[[Category:Functional programming]]

Latest revision as of 11:04, 1 February 2014

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 𝒞 is a category, and

F:𝒞𝒞

is an endofunctor of 𝒞, then an F-algebra is an object A of 𝒞 together with a 𝒞-morphism

α:F(A)A.

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

A homomorphism from an F-algebra (A,α) to an F-algebra (B,β) is a 𝒞-morphism

f:AB

such that

fα=βF(f),

see picture.

Thus the F-algebras constitute a category.

Example

Consider the functor F:SetSet 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 [zero,succ]:1+, which is the coproduct of the functions zero:1 (whose image is 0) and succ: (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 (,[zero,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.[1]

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.[1] 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.[2]

See also

Notes

  1. 1.0 1.1 Philip Wadler: Recursive types for free! University of Glasgow, June 1990. Draft.
  2. Robin Cockett: Charitable Thoughts (ps and ps.gz)

Further reading

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