What is this signature { zero : α; append : α -> β -> α; } name?

Question

I have Monoid and Builder interfaces:

interface IMonoidFor<T>
{
    T Zero { get; }
    T Append(T a, T b);
}

interface IBuilderFor<T>
{
    IBuilderFor<T> Append(T value);
    T Result();
}

And want to implement from Monoid to Builder converting algorithm. I tried to do this, and created something like this:

class BuilderMakerFor<T>
{
    public IBuilderFor<T> MakeFrom(IMonoidFor<T> monoid)
    {
        return new Product(monoid);
    }

    class Product
        : IBuilderFor<T>
    {
        readonly IMonoidFor<T> _monoid;
        T _result;

        public Product(
            IMonoidFor<T> monoid)
        {
            _monoid = monoid;
            _result = _monoid.Zero;
        }

        public IBuilderFor<T> Append(T value)
        {
            _result = _monoid.Append(_result, value);
            return this;
        }

        public T Result()
        {
            return _result;
        }
    }
}

But in real task, I need Builder that can accumulate items to list of this items, and others Builders with same principle of accumulation. My Builders must realize the next interface:

interface IBuilderFor<TSource, TResult>
{
    IBuilderFor<TSource, TResult> Append(TSource value);
    TResult Result();
}

And I need to have possibility to create Builders from { zero, append } signature. Like this:

interface X<TSource, TResult>
{
    TResult Zero { get; }
    TResult Append(TResult a, TSource b);
}

I implemented next BuilderMakerFor for new situation:

class BuilderMakerFor<TSource, TResult>
{
    public IBuilderFor<TSource, TResult> MakeFrom(X<TSource, TResult> x)
    {
        return new Product(x);
    }

    class Product
        : IBuilderFor<TSource, TResult>
    {
        readonly X<TSource, TResult> _x;
        TResult _result;

        public Product(
            X<TSource, TResult> x)
        {
            _x = x;
            _result = _x.Zero;
        }

        public IBuilderFor<TSource, TResult> Append(TSource value)
        {
            _result = _x.Append(_result, value);
            return this;
        }

        public TResult Result()
        {
            return _result;
        }
    }
}

All good, except one point: I don't know how to name X.

zero : α
append : α -> α -> α

name is Monoid.

What is

zero : α
append : α -> β -> α

name?

Answer 1

No answer during long period. And I think signature

zero : α
append : α -> β -> α

has no name in math.

I named it AggregatingSource or FoldingSource.