Algebraic structures in C++
In this repository, I'm playing around with my implementations of different algebraic structures in C++.
Structures
Semigroup
Semigroup is a pair (S, *) consisting of a set S together with an associative binary operation *: S -> S -> S. Note that for some type T all functions of type T -> S form a semigroup with an operation induced by the operation on semigroup:
f, g : T -> S;
t : T;
(f * g)(t) = f(t) * g(t).
This functionality is implemented using SemigroupFunctionLift class.
Monoid
Monoid is a pair (M, *) consisting of a set M together with an associative operation *: M -> M -> M. The pair must satisfy semigroup axioms and additionally M must contain a neutral element id such that:
(∀ a : M) a * id = id * a = a.
As with semigroups, functions to a monoid do form a monoid with the operation identical to operation on functions to semigroups. The neutral element is a constant function that maps all values to id: M.
When we have a container of monoids, we can apply a mconcat function that will collapse all the elements into one:
mconcat (a1, a2, ..., an) = id * a1 * a2 * ... * an.
Examples
Fizz Buzz using C++
We can solve the classical Fizz Buzz problem in a neat way using monoids.
Notice that strings together with an operation of concatenation with a separator between non-empty strings form a semigroup. Additionally, the empty string is a neutral element, so strings with such operation form a monoid. That means that the functions int -> string form monoid as well.
That means that we can implement a solution to a Fizz Buzz problem by defining each case as an instance of MonoidFunctionLift, throwing them into a container, and applying mconcat on a container of such functions. After that, we need to wrap everything up into a function that would replace the empty string with an original number and apply this function to all elements from 1 to 100.
This approach allows for great extensibility – we only need to add a new predicate function into a container and it would work.