represents a abstract machine with one state or a number of state. C# use state machine a lot. For example:

As mentioned in a previous part, in purely functional programming, functions cannot have side effects. For example, when defining LINQ queries, laziness and purity are expected. So, how should the imp

Theoretically, Tuple<> should be counted as the Id<> monad. However, it is lack of laziness. In the context of C# and LINQ, it is only monad-like.

Again, Lazy<> is the simplest monad, it is just the lazy version of Tuple<>, and should be considered as the Id<> monad. This is the implementation of its SelectMany:

Previous part introduced SelectMany for monad IEnumerable<>. Actually SelectMany is more than meets the eye, and can be used to implement other LINQ queries.

A previous part showed endofunctor category is a monoid (the entire category itself). An endofunctor In the endofunctor category can be monoid too. This kind of endofunctor is called monad. Formally,

Theoretically, Tuple<> should be counted as the Id<> monoidal functor. However, as previously mentioned, it is lack of laziness.

Lazy<> should be the simplest monoid functor - it is just the lazy version of Tuple<>. And in these posts it will be considered as the Id<> monoidal functor.

Given monoidal categories (C, ⊗, IC) and (D, ⊛, ID), a (or lax monoidal functors) is a functor F: C → D equipped with:

A previous part demonstrated endofunctor category is monoidal. Now with the help of bifunctor, the general abstract can be defined.