In mathematics, specifically in category theory, F-algebras generalize the notion of 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 functorF, 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 initialF-algebras which may serve to encapsulate the induction principle, and the dual construction F-coalgebras.
Definition
If C is a category, and F: C → C is an endofunctor of C, then an F-algebra is a tuple, where A is an object of C and α is a C-morphism F → A. The object A is called the 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 to an F-algebra is a C-morphism f: A→B such that f ∘ α = β ∘ F, according to the following commutative 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.
Examples
Groups
Classically, a group is a setG with a binary operation m : G × G → G, satisfying three axioms: Note that the closure axiom is included in the symbolic definition of m. To treat this in a categorical framework, we first define the identity and inverse as morphisms e and i respectively. Let C be an arbitrary category with finiteproducts and a terminal object *. The group G is an object in C. The morphism e sends each element in * to 1, the identity element of the group G. The morphism i sends each element x in G to its inverse x−1, satisfying m = m = 1. Then a group G can be defined as a 4-tuple, which describes a monoid category with only one object G. Each morphism f in this monoid category has an inverse f−1 that satisfies f−1 ∘ f = f ∘ f−1 = Id. It is then possible to rewrite the axioms in terms of morphisms: Then remove references to the elements of G : Which is the same as claiming commutativity for the following diagrams:
Now use the coproduct to glue the three morphisms in one: α = e + i + m according to This defines a group as a F-algebra where F is the functor F = 1 + G + G × G. Note 1: The above construction is used to define group objects over an arbitrary category with finite products and a terminal object *. When the category admits finite coproducts, the group objects are F-algebras. For example, finite groups are F-algebras in the category of finite sets and Lie groups are F-algebras in the category of smooth manifolds with smooth maps.
Algebraic structures
Going one step ahead of universal algebra, most algebraic structures are F-algebras. For example, abelian groups are F-algebras for the same functor F = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t = is the transpose on GxG. Monoids are F-algebras of signature F = 1 + M×M. In the same vein, semigroups are F-algebras of signature F = S×S Rings, domains and fields are also F-algebras with a signature involving two laws +,•: R×R → R, an additive identity 0: 1 → R, a multiplicative identity 1: 1 → R, and an additive inverse for each element -: R → R. As all these functions share the same codomainR they can be glued into a single signature function 1 + 1 + R + R×R + R×R → R, with axioms to express associativity, distributivity, and so on. This makes rings F-algebras on the category of sets with signature 1 + 1 + R + R×R + R×R. Alternatively, we can look at the functor F = 1 + R×R in the category of abelian groups. In that context, the multiplication is a homomorphism, meaning m = m + m and m = m + m, which are precisely the distributivity conditions. Therefore, a ring is an F-algebra of signature 1 + R×R over the category of abelian groups which satisfies two axioms. When we come to vector spaces and modules, the signature functor includes a scalar multiplicationk×E → E, and the signature F = 1 + E + k×E is parametrized by k over the category of fields, or rings. Algebras over a field can be viewed as F-algebras of signature 1 + 1 + A + A×A + A×A + k×A over the category of sets, of signature 1 + A×A over the category of modules, and of signature k×A over the category of rings, when they are associative and unitary.
Lattice
Not all mathematical structures are F-algebras. For example, a posetP may be defined in categorical terms with a morphism s:P × P → Ω, on a subobject classifier. The axioms restricting the morphism s to define a poset can be rewritten in terms of morphisms. However, as the codomain of s is Ω and not P, it is not an F-algebra. However, lattices in which every two elements have a supremum and an infimum, and in particular total orders, are F-algebras. This is because they can equivalently be defined in terms of the algebraic operations: x∨y = inf and x∧y = sup, subject to certain axioms. Thus they are F-algebras of signature P x P + P x P. It is often said that lattice theory draws on both order theory and universal algebra.
Recurrence
Consider the functor that sends a set to. Here, denotes the category of sets, denotes the usual coproduct given by the disjoint union, and is a terminal object. Then, the set of natural numbers together with the function —which is the coproduct of the functions and —is an F-algebra.
Initial ''F''-algebra
If the category of F-algebras for a given endofunctor F has an initial object, it is called an initial algebra. The algebra 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 alsoUniversal algebra.
