banana algebra

Abstract: I’ll be talking about the F-Algebra Category and related applications of F-Algebra in function programming in this post.

F-Algebra

So first of all, an algebra over a type \(a\) is set of functions that converts an algebra structure \(f a\) to \(a\) CoAlg. An algebra consists of:

• An algebra structure: \(\rm{F}\)
• A carrier type: \(\rm{a}\)
• A total function: \(\rm{F(a)} \to \rm{a}\)

An example of an algebra looks like:

• Algebra struct: data F1 a = Zero | One | Plus a a
• A carrier type: it could be any instance of a: Int, String, etc.
• A total function:

f1 :: F1 Int -> Int
f1 Zero       = 0
f1 One        = 1
f1 (Plus a b) = a + b

Or we can have:

f1' :: F1 String -> String
f1' Zero       = ""
f1' One        = "1"
f1' (Plus a b) = a ++ b

F-Algebra Arrows

All algebras for an algebra structure \(\rm{F}\) forms a category \(\cal{C}\). The objects are, of course, the algebras, while the arrows are defined as morphisms between each two pair of algebras that transforming the carrier type: \(\hom_{\cal{C}}(\rm{Alg}(\rm{F},\rm{a}), \rm{Alg}(\rm{F},\rm{b}))\).

       Alg(F,a)
F a --------------> a
          |
          |
          | <- hom(Alg(F,a), Alg(F,b))
          |
          v
F b --------------> b
      Alg(F,b)

For an arrow in F-algebra category, we need a transformation from F a to a.

References